{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0001","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d5cd5b13fbcbc9d4627e354b3d13eb24b4b3129c613434fe995b37e8bdc6273","family_id":"qf-mod28-ideal-qualification","seed":202610282001,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,1,2,4,6]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 2, 4, 6], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 4, 2, 0, 6, 4, 2], [0, 7, 6, 5, 4, 3, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 1, 2, 4, 6], 0, No, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], No)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 6], 0, No, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], No)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0002","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c13bbed77970591bdf072174f39d13d1048dd2637ebbd2a07362d5e77a9e67f5","family_id":"qf-mod28-ideal-qualification","seed":202610282002,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 4], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0003","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48a34c4bb64efbffaf84da6f2a3a2dccb99391648a63a8ed5c4c000fc5ae6cc7","family_id":"qf-mod28-ideal-qualification","seed":202610282003,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0004","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c40dadbf23aecc7d79915f05a20e41639fec2946805573f6df211dd32090def","family_id":"qf-mod28-ideal-qualification","seed":202610282004,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 0, 7, 5], [2, 3, 4, 6, 7, 1, 5, 0], [3, 4, 6, 7, 5, 2, 0, 1], [4, 6, 7, 5, 0, 3, 1, 2], [5, 0, 1, 2, 3, 7, 4, 6], [6, 7, 5, 0, 1, 4, 2, 3], [7, 5, 0, 1, 2, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 7, 2, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 7, 6, 4, 1, 3, 2], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 4, 2, 0, 2, 7, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0005","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49b3ab5d6b9afd4b29ea1d04d60fbf93b5eaf01b624815612dec079c4ce7d7e8","family_id":"qf-mod28-ideal-qualification","seed":202610282005,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 6, 0, 5], [2, 3, 4, 7, 5, 0, 1, 6], [3, 4, 7, 5, 6, 1, 2, 0], [4, 7, 5, 6, 0, 2, 3, 1], [5, 6, 0, 1, 2, 4, 7, 3], [6, 0, 1, 2, 3, 7, 5, 4], [7, 5, 6, 0, 1, 3, 4, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 5, 2], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 2, 5], [0, 6, 5, 7, 4, 2, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 4], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0006","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d1fb72e9114eb9a0b436a5b5544eb9742afe548d704e915a9d9b5d81f91ecb","family_id":"qf-mod28-ideal-qualification","seed":202610282006,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,1,2,4,6]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 0, 5, 6], [2, 3, 4, 7, 6, 1, 0, 5], [3, 4, 7, 6, 5, 2, 1, 0], [4, 7, 6, 5, 0, 3, 2, 1], [5, 0, 1, 2, 3, 6, 7, 4], [6, 5, 0, 1, 2, 7, 4, 3], [7, 6, 5, 0, 1, 4, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 6, 4, 2], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 6, 7, 4, 1, 2, 3], [0, 6, 4, 2, 0, 2, 4, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 1, 2, 4, 6], 0, No, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], No)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 6], 0, No, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], No)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0007","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6029c3869b6b3f70c694a7efc1622180f23a90b94829e3c5b5d90e60aae6def","family_id":"qf-mod28-ideal-qualification","seed":202610282007,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 5], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0008","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fbb1b50e99ff268db57fec0438271fae5f008391a53b1f283c68c23be43d29b","family_id":"qf-mod28-ideal-qualification","seed":202610282008,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 4, 0, 6], [2, 3, 5, 4, 6, 7, 1, 0], [3, 5, 4, 7, 0, 6, 2, 1], [4, 7, 6, 0, 2, 1, 5, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 0, 1, 2, 5, 3, 7, 4], [7, 6, 0, 1, 3, 2, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 2, 0, 7, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 7, 4, 3, 5, 1, 2], [0, 7, 5, 2, 7, 0, 2, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0009","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a01809aa83d91194156365a8a1e1d7e3fb358f5426cdc311383fb22d459cc173","family_id":"qf-mod28-ideal-qualification","seed":202610282009,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 6, 4, 0], [2, 3, 5, 6, 0, 4, 7, 1], [3, 5, 6, 4, 1, 7, 0, 2], [4, 7, 0, 1, 5, 2, 3, 6], [5, 6, 4, 7, 2, 0, 1, 3], [6, 4, 7, 0, 3, 1, 2, 5], [7, 0, 1, 2, 6, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 2, 4], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 4, 2], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 4, 6, 2, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 5], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0010","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"558ef4c0e44436ffd56860a7ca0ebebf22dfe8562dede53109fba9d38251bac6","family_id":"qf-mod28-ideal-qualification","seed":202610282010,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 6, 7, 4], [2, 3, 5, 6, 1, 7, 4, 0], [3, 5, 6, 7, 2, 4, 0, 1], [4, 0, 1, 2, 7, 3, 5, 6], [5, 6, 7, 4, 3, 0, 1, 2], [6, 7, 4, 0, 5, 1, 2, 3], [7, 4, 0, 1, 6, 2, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 7, 0, 2, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 7, 6, 1, 5, 3, 2], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 5, 2, 2, 0, 7, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 5], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0011","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff6470667634195bbcac18c4d2b57907f7f9d777eea0ccf78ef42158d8d00c4d","family_id":"qf-mod28-ideal-qualification","seed":202610282011,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,1,2,4,5]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 7, 0, 4], [2, 3, 5, 7, 0, 4, 1, 6], [3, 5, 7, 4, 1, 6, 2, 0], [4, 6, 0, 1, 5, 2, 7, 3], [5, 7, 4, 6, 2, 0, 3, 1], [6, 0, 1, 2, 7, 3, 4, 5], [7, 4, 6, 0, 3, 1, 5, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 4, 2], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 2, 4], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 4, 7, 2, 5, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 1, 2, 4, 5], 0, No, [[3, 1, 3], [6, 1, 6], [7, 1, 7]], [[1, 3, 3], [1, 6, 6], [1, 7, 7]], No)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 5], 0, No, [[3, 1, 3], [6, 1, 6], [7, 1, 7]], [[1, 3, 3], [1, 6, 6], [1, 7, 7]], No)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0012","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb23b00d38d1cb8e09b5741f626d2bb3dff281713f0e35337f8f0d43925317c2","family_id":"qf-mod28-ideal-qualification","seed":202610282012,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 7, 4, 6], [2, 3, 5, 7, 1, 6, 0, 4], [3, 5, 7, 6, 2, 4, 1, 0], [4, 0, 1, 2, 6, 3, 7, 5], [5, 7, 6, 4, 3, 0, 2, 1], [6, 4, 0, 1, 7, 2, 5, 3], [7, 6, 4, 0, 5, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 6, 0, 5, 2], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 6, 7, 1, 5, 2, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 2, 0, 5, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 5], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0013","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96029f377e23d37b4e2c30b971804edec772388d8c7969c8d49b1537ddb2fb9f","family_id":"qf-mod28-ideal-qualification","seed":202610282013,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 7, 4, 0], [2, 3, 6, 4, 7, 0, 5, 1], [3, 6, 4, 5, 0, 1, 7, 2], [4, 5, 7, 0, 2, 3, 1, 6], [5, 7, 0, 1, 3, 6, 2, 4], [6, 4, 5, 7, 1, 2, 0, 3], [7, 0, 1, 2, 6, 4, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 2, 6, 0, 5], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 6, 2, 5, 6, 0, 2], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 5, 4, 3, 2, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0014","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e203dcda660b5086229016ed0f190e3b3a20829867c582e312bdf3bcaf7d07cf","family_id":"qf-mod28-ideal-qualification","seed":202610282014,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 0, 4, 5], [2, 3, 6, 4, 5, 1, 7, 0], [3, 6, 4, 7, 0, 2, 5, 1], [4, 7, 5, 0, 2, 6, 1, 3], [5, 0, 1, 2, 6, 7, 3, 4], [6, 4, 7, 5, 1, 3, 0, 2], [7, 5, 0, 1, 3, 4, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 2, 7, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 7, 4, 3, 1, 6, 2], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 7, 2, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 6, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0015","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84e9a9ffb72d4c8e6e3a39dbb6c97c6a9a8086b6456d526d01f16a64885b918e","family_id":"qf-mod28-ideal-qualification","seed":202610282015,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 4, 5, 0], [2, 3, 6, 5, 0, 7, 4, 1], [3, 6, 5, 4, 1, 0, 7, 2], [4, 7, 0, 1, 6, 3, 2, 5], [5, 4, 7, 0, 3, 2, 1, 6], [6, 5, 4, 7, 2, 1, 0, 3], [7, 0, 1, 2, 5, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 2, 0, 4], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 4, 0, 2], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 4, 5, 2, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 6], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0016","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6393d4ad26acac5bb9919988bc17db513c5cfbb952bd987aa6d7fd02c7dce5af","family_id":"qf-mod28-ideal-qualification","seed":202610282016,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,1,2,6,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 7, 5, 4], [2, 3, 6, 5, 1, 4, 7, 0], [3, 6, 5, 7, 2, 0, 4, 1], [4, 0, 1, 2, 7, 6, 3, 5], [5, 7, 4, 0, 6, 2, 1, 3], [6, 5, 7, 4, 3, 1, 0, 2], [7, 4, 0, 1, 5, 3, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 7, 2, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 7, 5, 1, 3, 6, 2], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 2, 7, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 1, 2, 6, 7], 0, No, [[3, 1, 3], [4, 1, 4], [5, 1, 5]], [[1, 3, 3], [1, 4, 4], [1, 5, 5]], No)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 6, 7], 0, No, [[3, 1, 3], [4, 1, 4], [5, 1, 5]], [[1, 3, 3], [1, 4, 4], [1, 5, 5]], No)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0017","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b43c642856b1a876a17a4a12fab1a4945e0f3cc17422cad941e6e6a9129e497","family_id":"qf-mod28-ideal-qualification","seed":202610282017,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 6], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0018","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb61b09eab169804fb87bbbc494fdea7842cb60d5396c2580b006a045d617da1","family_id":"qf-mod28-ideal-qualification","seed":202610282018,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0019","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d43a8bc1019d4a11b3bebc2950c7290b4d044a2d9b11e00bc90cc79353285ee","family_id":"qf-mod28-ideal-qualification","seed":202610282019,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 6, 0, 4], [2, 3, 7, 4, 6, 0, 1, 5], [3, 7, 4, 5, 0, 1, 2, 6], [4, 5, 6, 0, 2, 3, 7, 1], [5, 6, 0, 1, 3, 7, 4, 2], [6, 0, 1, 2, 7, 4, 5, 3], [7, 4, 5, 6, 1, 2, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 2, 7, 5, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 7, 2, 5, 7, 2, 0], [0, 6, 5, 4, 3, 2, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0020","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"334512d2ba99e59d1e99db4d35127f4fed121899fa240772e37803ce154eb8c1","family_id":"qf-mod28-ideal-qualification","seed":202610282020,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 0, 5, 4], [2, 3, 7, 4, 5, 1, 0, 6], [3, 7, 4, 6, 0, 2, 1, 5], [4, 6, 5, 0, 2, 7, 3, 1], [5, 0, 1, 2, 7, 6, 4, 3], [6, 5, 0, 1, 3, 4, 7, 2], [7, 4, 6, 5, 1, 3, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 2, 6, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 6, 4, 3, 1, 2, 7], [0, 6, 7, 2, 6, 2, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0021","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3018dd5e0f218d7a98b06ffa0a88d22a630ea7a2c410650e69712d6a275ea2ae","family_id":"qf-mod28-ideal-qualification","seed":202610282021,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,1,2,4,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 1, 2, 4, 7], 0, No, [[3, 1, 3], [5, 1, 5], [6, 1, 6]], [[1, 3, 3], [1, 5, 5], [1, 6, 6]], No)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 4, 7], 0, No, [[3, 1, 3], [5, 1, 5], [6, 1, 6]], [[1, 3, 3], [1, 5, 5], [1, 6, 6]], No)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0022","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5778ef6f38ecba87cbcb3f2291ce78b43d6d85cff9cc61853384d886e520be4d","family_id":"qf-mod28-ideal-qualification","seed":202610282022,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0023","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9340de80197d9aea368447aa0b53b8ff3e90d944d37cf591db8d5974b87f9fff","family_id":"qf-mod28-ideal-qualification","seed":202610282023,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 0, 4, 6], [2, 3, 7, 6, 0, 1, 5, 4], [3, 7, 6, 4, 1, 2, 0, 5], [4, 5, 0, 1, 7, 6, 3, 2], [5, 0, 1, 2, 6, 4, 7, 3], [6, 4, 5, 0, 3, 7, 2, 1], [7, 6, 4, 5, 2, 3, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 4, 2, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 2, 4, 0], [0, 5, 4, 6, 2, 1, 3, 7], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0024","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be8fa5ca22af5d298fcf096a152e7fc55629df75cfdcc14f062556048bbf3f0a","family_id":"qf-mod28-ideal-qualification","seed":202610282024,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 4, 5, 6], [2, 3, 7, 6, 1, 0, 4, 5], [3, 7, 6, 5, 2, 1, 0, 4], [4, 0, 1, 2, 5, 6, 7, 3], [5, 4, 0, 1, 6, 7, 3, 2], [6, 5, 4, 0, 7, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 5, 7, 2, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 5, 6, 1, 2, 3, 7], [0, 5, 7, 2, 2, 7, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0025","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d65d3d704ff4401a7143ac3c0cc259bc8abb57bea6488a95145c526477405d06","family_id":"qf-mod28-ideal-qualification","seed":202610282025,"semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"},"fields":{"title":"Two-Sided Ideals Require Three Complete Qualifications","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 4, 5, 3, 6, 7, 0], [2, 4, 3, 6, 5, 7, 0, 1], [3, 5, 6, 0, 7, 1, 2, 4], [4, 3, 5, 7, 6, 0, 1, 2], [5, 6, 7, 1, 0, 2, 4, 3], [6, 7, 0, 2, 1, 4, 3, 5], [7, 0, 1, 4, 2, 3, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 3, 0, 6, 2, 3, 6], [0, 3, 0, 0, 3, 3, 0, 3], [0, 4, 6, 3, 1, 7, 2, 5], [0, 5, 2, 3, 7, 1, 6, 4], [0, 6, 3, 0, 2, 6, 3, 2], [0, 7, 6, 3, 5, 4, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 3] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.","answer":{"kind":"expression","value":"([0, 3], 0, Yes, [], [], Yes)","explanation":"Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 3], 0, Yes, [], [], Yes)."},"skills":["Two-Sided Ideals Require Three Complete Qualifications"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0026","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c4cff24b94f695e47a36f8b3c8c598c1281f65b5cb06b2ad446e25134de681","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282026,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,2,4,6]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 2, 4, 6], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 4, 2, 0, 6, 4, 2], [0, 7, 6, 5, 4, 3, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 6], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 6], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0027","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2708b35188291ad35fc666caf0857b015f7e72f94da326e64f6df92e9a6a33f6","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282027,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 4], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0028","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f8848412309f09a49d1f7c4cfed715a231fb0e079648b5d7c87793895d22c36","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282028,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0029","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d975d956012680590c117da60a75ebea9cfdfc2d35db41e43b08bd6919190547","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282029,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 0, 7, 5], [2, 3, 4, 6, 7, 1, 5, 0], [3, 4, 6, 7, 5, 2, 0, 1], [4, 6, 7, 5, 0, 3, 1, 2], [5, 0, 1, 2, 3, 7, 4, 6], [6, 7, 5, 0, 1, 4, 2, 3], [7, 5, 0, 1, 2, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 7, 2, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 7, 6, 4, 1, 3, 2], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 4, 2, 0, 2, 7, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0030","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4843c25dcd687f3343f3bc7f96e85b644e68eb3e2cc7768bb1d8418d9e01486d","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282030,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 6, 0, 5], [2, 3, 4, 7, 5, 0, 1, 6], [3, 4, 7, 5, 6, 1, 2, 0], [4, 7, 5, 6, 0, 2, 3, 1], [5, 6, 0, 1, 2, 4, 7, 3], [6, 0, 1, 2, 3, 7, 5, 4], [7, 5, 6, 0, 1, 3, 4, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 5, 2], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 2, 5], [0, 6, 5, 7, 4, 2, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 4], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0031","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c91386f4f928f1572f540e0ae6fc92bf293dd4de80000f27f09e8c8637090fa","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282031,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,2,4,6]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 0, 5, 6], [2, 3, 4, 7, 6, 1, 0, 5], [3, 4, 7, 6, 5, 2, 1, 0], [4, 7, 6, 5, 0, 3, 2, 1], [5, 0, 1, 2, 3, 6, 7, 4], [6, 5, 0, 1, 2, 7, 4, 3], [7, 6, 5, 0, 1, 4, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 6, 4, 2], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 6, 7, 4, 1, 2, 3], [0, 6, 4, 2, 0, 2, 4, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 6], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 6], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0032","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb703081210ee69b6b635160d1279fb0aa77c121f49970f127907a2d90b6bacd","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282032,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 5], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0033","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23c1bfa4502f4b7ad45d0f68d0c85cf65772b52a948cd20ec2c010e1c71e55ba","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282033,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 4, 0, 6], [2, 3, 5, 4, 6, 7, 1, 0], [3, 5, 4, 7, 0, 6, 2, 1], [4, 7, 6, 0, 2, 1, 5, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 0, 1, 2, 5, 3, 7, 4], [7, 6, 0, 1, 3, 2, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 2, 0, 7, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 7, 4, 3, 5, 1, 2], [0, 7, 5, 2, 7, 0, 2, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0034","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fce670da1e04894ff02a3373939d312d56aa24c98cf91110803d895869e0d4b","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282034,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 6, 4, 0], [2, 3, 5, 6, 0, 4, 7, 1], [3, 5, 6, 4, 1, 7, 0, 2], [4, 7, 0, 1, 5, 2, 3, 6], [5, 6, 4, 7, 2, 0, 1, 3], [6, 4, 7, 0, 3, 1, 2, 5], [7, 0, 1, 2, 6, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 2, 4], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 4, 2], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 4, 6, 2, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 5], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0035","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a68c1c52563088f4fda6f7d7a61aeb98ac226e42c154e860e290aa627ebd642","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282035,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 6, 7, 4], [2, 3, 5, 6, 1, 7, 4, 0], [3, 5, 6, 7, 2, 4, 0, 1], [4, 0, 1, 2, 7, 3, 5, 6], [5, 6, 7, 4, 3, 0, 1, 2], [6, 7, 4, 0, 5, 1, 2, 3], [7, 4, 0, 1, 6, 2, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 7, 0, 2, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 7, 6, 1, 5, 3, 2], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 5, 2, 2, 0, 7, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 5], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0036","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8eacfe311574296379d8914cc8d4bd116a5c5d08420790f93badf10861258e","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282036,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 7, 0, 4], [2, 3, 5, 7, 0, 4, 1, 6], [3, 5, 7, 4, 1, 6, 2, 0], [4, 6, 0, 1, 5, 2, 7, 3], [5, 7, 4, 6, 2, 0, 3, 1], [6, 0, 1, 2, 7, 3, 4, 5], [7, 4, 6, 0, 3, 1, 5, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 4, 2], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 2, 4], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 4, 7, 2, 5, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 5], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0037","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"282e557111c334e345806fcd2602486d708e13ff69e9afd0ddd1e2638e3dcaca","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282037,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 7, 4, 6], [2, 3, 5, 7, 1, 6, 0, 4], [3, 5, 7, 6, 2, 4, 1, 0], [4, 0, 1, 2, 6, 3, 7, 5], [5, 7, 6, 4, 3, 0, 2, 1], [6, 4, 0, 1, 7, 2, 5, 3], [7, 6, 4, 0, 5, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 6, 0, 5, 2], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 6, 7, 1, 5, 2, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 2, 0, 5, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 5], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0038","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e74fb58ac8ac8dfbd057863b0297f92a3b4e953ed05f78b71f7aaf588093107a","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282038,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 7, 4, 0], [2, 3, 6, 4, 7, 0, 5, 1], [3, 6, 4, 5, 0, 1, 7, 2], [4, 5, 7, 0, 2, 3, 1, 6], [5, 7, 0, 1, 3, 6, 2, 4], [6, 4, 5, 7, 1, 2, 0, 3], [7, 0, 1, 2, 6, 4, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 2, 6, 0, 5], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 6, 2, 5, 6, 0, 2], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 5, 4, 3, 2, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0039","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"036a2cb90e6106c52f90f1a0f6cd8b1120bdcd87d0ccc81944ade5d88bc3da78","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282039,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 0, 4, 5], [2, 3, 6, 4, 5, 1, 7, 0], [3, 6, 4, 7, 0, 2, 5, 1], [4, 7, 5, 0, 2, 6, 1, 3], [5, 0, 1, 2, 6, 7, 3, 4], [6, 4, 7, 5, 1, 3, 0, 2], [7, 5, 0, 1, 3, 4, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 2, 7, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 7, 4, 3, 1, 6, 2], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 7, 2, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 6, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0040","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a248a41efc354c9877cd2580e4fe107dbe207df50ad7e71dd642be65d51b8772","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282040,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 4, 5, 0], [2, 3, 6, 5, 0, 7, 4, 1], [3, 6, 5, 4, 1, 0, 7, 2], [4, 7, 0, 1, 6, 3, 2, 5], [5, 4, 7, 0, 3, 2, 1, 6], [6, 5, 4, 7, 2, 1, 0, 3], [7, 0, 1, 2, 5, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 2, 0, 4], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 4, 0, 2], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 4, 5, 2, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 6], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0041","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5d3346c3539e8b08090b58fd408af902d5248396758a95cf1ebac7e6fa84204","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282041,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 7, 5, 4], [2, 3, 6, 5, 1, 4, 7, 0], [3, 6, 5, 7, 2, 0, 4, 1], [4, 0, 1, 2, 7, 6, 3, 5], [5, 7, 4, 0, 6, 2, 1, 3], [6, 5, 7, 4, 3, 1, 0, 2], [7, 4, 0, 1, 5, 3, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 7, 2, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 7, 5, 1, 3, 6, 2], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 2, 7, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 6, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0042","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d280fe16314303adc301bbb3574a885feed3e8dbf13ca3d6c3120fca7663303","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282042,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 6], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0043","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ca8f9db9b8eb5065ec8575fdf552c65d54ac46a757200395b487e04aa295ee","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282043,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0044","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50cd130a6e8b5936f4d32ed003ee51262a5436cfa9c80c5565cb495bc3472177","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282044,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 6, 0, 4], [2, 3, 7, 4, 6, 0, 1, 5], [3, 7, 4, 5, 0, 1, 2, 6], [4, 5, 6, 0, 2, 3, 7, 1], [5, 6, 0, 1, 3, 7, 4, 2], [6, 0, 1, 2, 7, 4, 5, 3], [7, 4, 5, 6, 1, 2, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 2, 7, 5, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 7, 2, 5, 7, 2, 0], [0, 6, 5, 4, 3, 2, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0045","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79af9faafbb7686a3e9c59baa4e5a4b1820b3075d667824c27a0535aadd82158","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282045,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 0, 5, 4], [2, 3, 7, 4, 5, 1, 0, 6], [3, 7, 4, 6, 0, 2, 1, 5], [4, 6, 5, 0, 2, 7, 3, 1], [5, 0, 1, 2, 7, 6, 4, 3], [6, 5, 0, 1, 3, 4, 7, 2], [7, 4, 6, 5, 1, 3, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 2, 6, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 6, 4, 3, 1, 2, 7], [0, 6, 7, 2, 6, 2, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0046","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4900e69cb1be4b4d83760dfcd69ad49f6739f456886bc4439c68b1bd5b88cb71","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282046,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,2,4,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0047","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b5e64e12a31a8e967651026ba1ff7396df141533d40d1bfa4e0681efc38856","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282047,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0048","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b54f3163b932f1fce5ae09673bf81d916b25fbcc87707fa6e5b913eea1ed155d","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282048,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 0, 4, 6], [2, 3, 7, 6, 0, 1, 5, 4], [3, 7, 6, 4, 1, 2, 0, 5], [4, 5, 0, 1, 7, 6, 3, 2], [5, 0, 1, 2, 6, 4, 7, 3], [6, 4, 5, 0, 3, 7, 2, 1], [7, 6, 4, 5, 2, 3, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 4, 2, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 2, 4, 0], [0, 5, 4, 6, 2, 1, 3, 7], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0049","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f08b560288ebf37a7e068364027f044abace20bbb4cef6c2c58325ee81488fa5","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282049,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 4, 5, 6], [2, 3, 7, 6, 1, 0, 4, 5], [3, 7, 6, 5, 2, 1, 0, 4], [4, 0, 1, 2, 5, 6, 7, 3], [5, 4, 0, 1, 6, 7, 3, 2], [6, 5, 4, 0, 7, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 5, 7, 2, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 5, 6, 1, 2, 3, 7], [0, 5, 7, 2, 2, 7, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0050","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af288dd9212befc8fbc8e61ccc85b07adbc22100fc504fd7bb336d09dd0a4e43","family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282050,"semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"},"fields":{"title":"Check the Additive Structure before Multiplicative Absorption","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 4, 5, 3, 6, 7, 0], [2, 4, 3, 6, 5, 7, 0, 1], [3, 5, 6, 0, 7, 1, 2, 4], [4, 3, 5, 7, 6, 0, 1, 2], [5, 6, 7, 1, 0, 2, 4, 3], [6, 7, 0, 2, 1, 4, 3, 5], [7, 0, 1, 4, 2, 3, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 3, 0, 6, 2, 3, 6], [0, 3, 0, 0, 3, 3, 0, 3], [0, 4, 6, 3, 1, 7, 2, 5], [0, 5, 2, 3, 7, 1, 6, 4], [0, 6, 3, 0, 2, 6, 3, 2], [0, 7, 6, 3, 5, 4, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 3] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,all missing [a,b,a+b] rows,all missing [a,additive_inverse] rows,additive_subgroup?).","answer":{"kind":"expression","value":"([0, 3], 0, [], [], Yes)","explanation":"Use the normalized subset order, testing every ordered addition pair and each additive inverse. The subset must also be nonempty and contain the actual additive zero. No absorption or ideal claim follows from this flag alone. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 3], 0, [], [], Yes)."},"skills":["Check the Additive Structure before Multiplicative Absorption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0051","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05f0c247b87b9e103beb27388e332da4e04371164f7d50fa572fbe8a245ef147","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282051,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,2,4,6]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 2, 4, 6], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 4, 2, 0, 6, 4, 2], [0, 7, 6, 5, 4, 3, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 6], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 6], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0052","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b84b3aea24c0d210bbad99b06a02712be548f05d5a400bd13fc064838c71748","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282052,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 4], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0053","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8557f7e4d985aa1d745a2c0929a42842b8df4f550ff2e3ae12ba3d55b95ad584","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282053,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0054","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efb49d0ac14a927662af0626fb0e46aa046b94824705a354d8f6e5a1015db18f","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282054,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 0, 7, 5], [2, 3, 4, 6, 7, 1, 5, 0], [3, 4, 6, 7, 5, 2, 0, 1], [4, 6, 7, 5, 0, 3, 1, 2], [5, 0, 1, 2, 3, 7, 4, 6], [6, 7, 5, 0, 1, 4, 2, 3], [7, 5, 0, 1, 2, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 7, 2, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 7, 6, 4, 1, 3, 2], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 4, 2, 0, 2, 7, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0055","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7c71a3b7f19508f0e97602cf1589f0a41604314101d71a122afad5c8393b9de","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282055,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 6, 0, 5], [2, 3, 4, 7, 5, 0, 1, 6], [3, 4, 7, 5, 6, 1, 2, 0], [4, 7, 5, 6, 0, 2, 3, 1], [5, 6, 0, 1, 2, 4, 7, 3], [6, 0, 1, 2, 3, 7, 5, 4], [7, 5, 6, 0, 1, 3, 4, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 5, 2], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 2, 5], [0, 6, 5, 7, 4, 2, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 4] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 4], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 4], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0056","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"691c9f2c040cd360c5d1df7a11a49f9b52f7a41a911c7826341cca824df4cc1b","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282056,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,2,4,6]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 0, 5, 6], [2, 3, 4, 7, 6, 1, 0, 5], [3, 4, 7, 6, 5, 2, 1, 0], [4, 7, 6, 5, 0, 3, 2, 1], [5, 0, 1, 2, 3, 6, 7, 4], [6, 5, 0, 1, 2, 7, 4, 3], [7, 6, 5, 0, 1, 4, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 6, 4, 2], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 6, 7, 4, 1, 2, 3], [0, 6, 4, 2, 0, 2, 4, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 6], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 6], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0057","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05af580fc9259b366a89a54b8d73123f7b2bd46933c0c75fe183377fb096988a","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282057,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 5], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0058","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad3e111126f8c4b2834b44547192e722cda18f466e8c7f84fc39d55e0453b6f4","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282058,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 4, 0, 6], [2, 3, 5, 4, 6, 7, 1, 0], [3, 5, 4, 7, 0, 6, 2, 1], [4, 7, 6, 0, 2, 1, 5, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 0, 1, 2, 5, 3, 7, 4], [7, 6, 0, 1, 3, 2, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 2, 0, 7, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 7, 4, 3, 5, 1, 2], [0, 7, 5, 2, 7, 0, 2, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0059","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b0974d1066d4c626896add1654b5df4e16d6665adfb6c4e8a367ca452fda631","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282059,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 6, 4, 0], [2, 3, 5, 6, 0, 4, 7, 1], [3, 5, 6, 4, 1, 7, 0, 2], [4, 7, 0, 1, 5, 2, 3, 6], [5, 6, 4, 7, 2, 0, 1, 3], [6, 4, 7, 0, 3, 1, 2, 5], [7, 0, 1, 2, 6, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 2, 4], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 4, 2], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 4, 6, 2, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 5], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 5], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0060","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b96b4eb2c48bef2b0340b963a3b6e4467e5a02fb30b15d106a5ca0eca4e11f27","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282060,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 6, 7, 4], [2, 3, 5, 6, 1, 7, 4, 0], [3, 5, 6, 7, 2, 4, 0, 1], [4, 0, 1, 2, 7, 3, 5, 6], [5, 6, 7, 4, 3, 0, 1, 2], [6, 7, 4, 0, 5, 1, 2, 3], [7, 4, 0, 1, 6, 2, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 7, 0, 2, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 7, 6, 1, 5, 3, 2], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 5, 2, 2, 0, 7, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 5], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0061","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e012acc62fc8bfe5d7facc0346d86b97e6918ecc89e5a383e1e66448bcac260","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282061,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 7, 0, 4], [2, 3, 5, 7, 0, 4, 1, 6], [3, 5, 7, 4, 1, 6, 2, 0], [4, 6, 0, 1, 5, 2, 7, 3], [5, 7, 4, 6, 2, 0, 3, 1], [6, 0, 1, 2, 7, 3, 4, 5], [7, 4, 6, 0, 3, 1, 5, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 4, 2], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 2, 4], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 4, 7, 2, 5, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 5], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 5], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0062","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df5dae7d5077c0fb38480fd05318ac585ad31d88a30e310b5ff69c8fcac0c6f3","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282062,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 7, 4, 6], [2, 3, 5, 7, 1, 6, 0, 4], [3, 5, 7, 6, 2, 4, 1, 0], [4, 0, 1, 2, 6, 3, 7, 5], [5, 7, 6, 4, 3, 0, 2, 1], [6, 4, 0, 1, 7, 2, 5, 3], [7, 6, 4, 0, 5, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 6, 0, 5, 2], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 6, 7, 1, 5, 2, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 2, 0, 5, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 5] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 5], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 5], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0063","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fcc17524658297413736d51d4df523888febe54dcc8b1b6071a20451386fa46","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282063,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 7, 4, 0], [2, 3, 6, 4, 7, 0, 5, 1], [3, 6, 4, 5, 0, 1, 7, 2], [4, 5, 7, 0, 2, 3, 1, 6], [5, 7, 0, 1, 3, 6, 2, 4], [6, 4, 5, 7, 1, 2, 0, 3], [7, 0, 1, 2, 6, 4, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 2, 6, 0, 5], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 6, 2, 5, 6, 0, 2], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 5, 4, 3, 2, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0064","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8b134d8287de9a4448578fdd6ee3c20b9e82fc7d0bf4e6b03fc5765b7f1635","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282064,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 0, 4, 5], [2, 3, 6, 4, 5, 1, 7, 0], [3, 6, 4, 7, 0, 2, 5, 1], [4, 7, 5, 0, 2, 6, 1, 3], [5, 0, 1, 2, 6, 7, 3, 4], [6, 4, 7, 5, 1, 3, 0, 2], [7, 5, 0, 1, 3, 4, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 2, 7, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 7, 4, 3, 1, 6, 2], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 7, 2, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 6, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 6, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0065","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f911dc8433d192b0c3a01ba7ab5efa6b5bb512ced6f693e27903f25e0e6d0ec1","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282065,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 4, 5, 0], [2, 3, 6, 5, 0, 7, 4, 1], [3, 6, 5, 4, 1, 0, 7, 2], [4, 7, 0, 1, 6, 3, 2, 5], [5, 4, 7, 0, 3, 2, 1, 6], [6, 5, 4, 7, 2, 1, 0, 3], [7, 0, 1, 2, 5, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 2, 0, 4], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 4, 0, 2], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 4, 5, 2, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 6], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0066","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b9ff47b62b698ca792332420ac53b64d41bd07d65ce5b4f7379acb5028932f","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282066,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 7, 5, 4], [2, 3, 6, 5, 1, 4, 7, 0], [3, 6, 5, 7, 2, 0, 4, 1], [4, 0, 1, 2, 7, 6, 3, 5], [5, 7, 4, 0, 6, 2, 1, 3], [6, 5, 7, 4, 3, 1, 0, 2], [7, 4, 0, 1, 5, 3, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 7, 2, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 7, 5, 1, 3, 6, 2], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 2, 7, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 6, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 6, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0067","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c54c9871bddc2d409f49f81514edc371a2cb33687669f14c1aa3ce0338e10de","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282067,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 6] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 6], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 6], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0068","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6afc66abf4caaced6398290e8e90cb2a80dcaa455118a3b9c5553b357d7e55fe","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282068,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0069","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b58658c34be98e5e67d75ded18439574b13a22defddf56d89d1bcdb0af51905","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282069,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 6, 0, 4], [2, 3, 7, 4, 6, 0, 1, 5], [3, 7, 4, 5, 0, 1, 2, 6], [4, 5, 6, 0, 2, 3, 7, 1], [5, 6, 0, 1, 3, 7, 4, 2], [6, 0, 1, 2, 7, 4, 5, 3], [7, 4, 5, 6, 1, 2, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 2, 7, 5, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 7, 2, 5, 7, 2, 0], [0, 6, 5, 4, 3, 2, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 5, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0070","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2024d421965ef3aba2efdfeea17f7ec0448a94eb8a89f1224d057c0b77fc60fa","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282070,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 0, 5, 4], [2, 3, 7, 4, 5, 1, 0, 6], [3, 7, 4, 6, 0, 2, 1, 5], [4, 6, 5, 0, 2, 7, 3, 1], [5, 0, 1, 2, 7, 6, 4, 3], [6, 5, 0, 1, 3, 4, 7, 2], [7, 4, 6, 5, 1, 3, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 2, 6, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 6, 4, 3, 1, 2, 7], [0, 6, 7, 2, 6, 2, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0071","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b70c20ff9be53a58340153d6e8ea07aadda5c544db746d7f875e6bbb8e0597c","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282071,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,2,4,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 4, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 4, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0072","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"027e44400f665f30d82ddbe407b1449b7610191244939e443df1298f911d9a0a","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282072,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0073","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af0a625931d8a4815261d9228ae3afae942b3911a0b8f827442c33d7ca15088f","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282073,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 0, 4, 6], [2, 3, 7, 6, 0, 1, 5, 4], [3, 7, 6, 4, 1, 2, 0, 5], [4, 5, 0, 1, 7, 6, 3, 2], [5, 0, 1, 2, 6, 4, 7, 3], [6, 4, 5, 0, 3, 7, 2, 1], [7, 6, 4, 5, 2, 3, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 4, 2, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 2, 4, 0], [0, 5, 4, 6, 2, 1, 3, 7], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0074","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16049776c6835993974479db3cf993e2693443570d03da64f12462d6b2fb3a8f","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282074,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 4, 5, 6], [2, 3, 7, 6, 1, 0, 4, 5], [3, 7, 6, 5, 2, 1, 0, 4], [4, 0, 1, 2, 5, 6, 7, 3], [5, 4, 0, 1, 6, 7, 3, 2], [6, 5, 4, 0, 7, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 5, 7, 2, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 5, 6, 1, 2, 3, 7], [0, 5, 7, 2, 2, 7, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 2, 5, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 2, 5, 7], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 2, 5, 7], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0075","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d4dab139c88dde2069a95ac08cfc7734ce185a3eefbc9b73dc3bb7143a0fb17","family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282075,"semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"},"fields":{"title":"Left Absorption and Right Absorption Must Both Be Reported","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 4, 5, 3, 6, 7, 0], [2, 4, 3, 6, 5, 7, 0, 1], [3, 5, 6, 0, 7, 1, 2, 4], [4, 3, 5, 7, 6, 0, 1, 2], [5, 6, 7, 1, 0, 2, 4, 3], [6, 7, 0, 2, 1, 4, 3, 5], [7, 0, 1, 4, 2, 3, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 3, 0, 6, 2, 3, 6], [0, 3, 0, 0, 3, 3, 0, 3], [0, 4, 6, 3, 1, 7, 2, 5], [0, 5, 2, 3, 7, 1, 6, 4], [0, 6, 3, 0, 2, 6, 3, 2], [0, 7, 6, 3, 5, 4, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 3] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.","answer":{"kind":"expression","value":"([0, 3], [], [], Yes, Yes)","explanation":"Test each product on both sides using all parent-ring elements, even in a noncommutative ring. An empty subset passes these universal absorption checks vacuously but fails additive-subgroup and ideal qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 3], [], [], Yes, Yes)."},"skills":["Left Absorption and Right Absorption Must Both Be Reported"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0076","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94dde32f1c6c98c34abeadb2016bb2af7852400171a19b60c4d7a4db055068a7","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282076,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 2, 4, 6], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 4, 2, 0, 6, 4, 2], [0, 7, 6, 5, 4, 3, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0077","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f71aee0d092fdf2a05d7b220ed1f313c6578721fb59099d005e97b81481d3cfc","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282077,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0078","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7adc0004ca66a9480e8643f810c38a3bca53a70869a6f8ddd9d877f451eb0cd","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282078,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0079","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba6b23f581dd62e702b461092f8d08511ebece69799ba5103b8ea3098d71fdf4","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282079,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 0, 7, 5], [2, 3, 4, 6, 7, 1, 5, 0], [3, 4, 6, 7, 5, 2, 0, 1], [4, 6, 7, 5, 0, 3, 1, 2], [5, 0, 1, 2, 3, 7, 4, 6], [6, 7, 5, 0, 1, 4, 2, 3], [7, 5, 0, 1, 2, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 7, 2, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 7, 6, 4, 1, 3, 2], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 4, 2, 0, 2, 7, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0080","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8795922ae44f1579bbb3bfc2a7e14dc1433ca742d6522b2a1613560fc5bec95e","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282080,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 6, 0, 5], [2, 3, 4, 7, 5, 0, 1, 6], [3, 4, 7, 5, 6, 1, 2, 0], [4, 7, 5, 6, 0, 2, 3, 1], [5, 6, 0, 1, 2, 4, 7, 3], [6, 0, 1, 2, 3, 7, 5, 4], [7, 5, 6, 0, 1, 3, 4, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 5, 2], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 2, 5], [0, 6, 5, 7, 4, 2, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0081","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc38af082fda7e047972d072d18b4235e1f7fa83a4d2147854130f9c4e8f2dfd","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282081,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 7, 0, 5, 6], [2, 3, 4, 7, 6, 1, 0, 5], [3, 4, 7, 6, 5, 2, 1, 0], [4, 7, 6, 5, 0, 3, 2, 1], [5, 0, 1, 2, 3, 6, 7, 4], [6, 5, 0, 1, 2, 7, 4, 3], [7, 6, 5, 0, 1, 4, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 6, 0, 6, 4, 2], [0, 3, 6, 1, 4, 7, 2, 5], [0, 4, 0, 4, 0, 4, 0, 4], [0, 5, 6, 7, 4, 1, 2, 3], [0, 6, 4, 2, 0, 2, 4, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0082","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71c8afcbba115745172a827b7e0b1acfee4ce823ca4ca1e8564de7457e9ecebf","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282082,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0083","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f7005837545a38fe288ab9d5ee2f37aa2766ff3a011199bdf528f2c513a4c43","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282083,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 4, 0, 6], [2, 3, 5, 4, 6, 7, 1, 0], [3, 5, 4, 7, 0, 6, 2, 1], [4, 7, 6, 0, 2, 1, 5, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 0, 1, 2, 5, 3, 7, 4], [7, 6, 0, 1, 3, 2, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 2, 0, 7, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 7, 4, 3, 5, 1, 2], [0, 7, 5, 2, 7, 0, 2, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0084","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b8160ed8678f82f814a74ff2e0334c0d3b81f651e1b2b74afe8ed67b70ee30d","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282084,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 7, 6, 4, 0], [2, 3, 5, 6, 0, 4, 7, 1], [3, 5, 6, 4, 1, 7, 0, 2], [4, 7, 0, 1, 5, 2, 3, 6], [5, 6, 4, 7, 2, 0, 1, 3], [6, 4, 7, 0, 3, 1, 2, 5], [7, 0, 1, 2, 6, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 2, 4], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 4, 2], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 4, 6, 2, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0085","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b9343f9edda06bdc43fa4bbf78ba4d6295a1dd17cff57a300472c3121808da","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282085,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 6, 7, 4], [2, 3, 5, 6, 1, 7, 4, 0], [3, 5, 6, 7, 2, 4, 0, 1], [4, 0, 1, 2, 7, 3, 5, 6], [5, 6, 7, 4, 3, 0, 1, 2], [6, 7, 4, 0, 5, 1, 2, 3], [7, 4, 0, 1, 6, 2, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 7, 7, 0, 2, 5], [0, 3, 7, 1, 6, 5, 4, 2], [0, 4, 7, 6, 1, 5, 3, 2], [0, 5, 0, 5, 5, 0, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 5, 2, 2, 0, 7, 5]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0086","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b428f72e9ac68f0141c421a1d42e34aefab0f56eed87f73628a6d3a53930c41c","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282086,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 7, 0, 4], [2, 3, 5, 7, 0, 4, 1, 6], [3, 5, 7, 4, 1, 6, 2, 0], [4, 6, 0, 1, 5, 2, 7, 3], [5, 7, 4, 6, 2, 0, 3, 1], [6, 0, 1, 2, 7, 3, 4, 5], [7, 4, 6, 0, 3, 1, 5, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 4, 5, 0, 4, 2], [0, 3, 4, 1, 2, 5, 7, 6], [0, 4, 5, 2, 5, 0, 2, 4], [0, 5, 0, 5, 0, 0, 5, 5], [0, 6, 4, 7, 2, 5, 1, 3], [0, 7, 2, 6, 4, 5, 3, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0087","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"681b11e0551512a21482f5de9293bfdb84234698063ff23e7b44ec03888d406f","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282087,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 0, 7, 4, 6], [2, 3, 5, 7, 1, 6, 0, 4], [3, 5, 7, 6, 2, 4, 1, 0], [4, 0, 1, 2, 6, 3, 7, 5], [5, 7, 6, 4, 3, 0, 2, 1], [6, 4, 0, 1, 7, 2, 5, 3], [7, 6, 4, 0, 5, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 6, 0, 5, 2], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 6, 7, 1, 5, 2, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 2, 0, 5, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0088","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f382bec08aa50d4839ddd5293e6e37887ef77e35b5c61d86e2ea02910e2645b","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282088,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 7, 4, 0], [2, 3, 6, 4, 7, 0, 5, 1], [3, 6, 4, 5, 0, 1, 7, 2], [4, 5, 7, 0, 2, 3, 1, 6], [5, 7, 0, 1, 3, 6, 2, 4], [6, 4, 5, 7, 1, 2, 0, 3], [7, 0, 1, 2, 6, 4, 3, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 2, 6, 0, 5], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 6, 2, 5, 6, 0, 2], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 5, 4, 3, 2, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0089","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c766f5987719de315730d9c506ec83309841a579c28bf76a99aee728e01638cc","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282089,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 0, 4, 5], [2, 3, 6, 4, 5, 1, 7, 0], [3, 6, 4, 7, 0, 2, 5, 1], [4, 7, 5, 0, 2, 6, 1, 3], [5, 0, 1, 2, 6, 7, 3, 4], [6, 4, 7, 5, 1, 3, 0, 2], [7, 5, 0, 1, 3, 4, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 2, 7, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 7, 4, 3, 1, 6, 2], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 7, 2, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0090","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bdac3cbd3c0c0f9aea2d64002b296b95477f6fd63203108a3406a2efb1075b2","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282090,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 7, 4, 5, 0], [2, 3, 6, 5, 0, 7, 4, 1], [3, 6, 5, 4, 1, 0, 7, 2], [4, 7, 0, 1, 6, 3, 2, 5], [5, 4, 7, 0, 3, 2, 1, 6], [6, 5, 4, 7, 2, 1, 0, 3], [7, 0, 1, 2, 5, 6, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 2, 0, 4], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 4, 0, 2], [0, 5, 2, 7, 4, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 4, 5, 2, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0091","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc78f64d4b4dd509650d770fe19a7d352203e51c574a5f3af70a9e3f45da341a","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282091,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 7, 5, 4], [2, 3, 6, 5, 1, 4, 7, 0], [3, 6, 5, 7, 2, 0, 4, 1], [4, 0, 1, 2, 7, 6, 3, 5], [5, 7, 4, 0, 6, 2, 1, 3], [6, 5, 7, 4, 3, 1, 0, 2], [7, 4, 0, 1, 5, 3, 2, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 7, 7, 2, 0, 6], [0, 3, 7, 1, 5, 4, 6, 2], [0, 4, 7, 5, 1, 3, 6, 2], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 0, 6, 6, 6, 0, 0], [0, 7, 6, 2, 2, 7, 0, 6]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0092","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8cf5f8a77bacd128c1e9f19bac4d4254feafc8d797d4d598e6ca092d6f42da3","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282092,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0093","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ae346896a607213dafcd127ff8852d2af3455ca99c9cdc75f8638f19459b792","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282093,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0094","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a15fe8ffdf6b3f0fb79d97af5baf9f77561ac4e5be86b4474dd6f76881abefa","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282094,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 6, 0, 4], [2, 3, 7, 4, 6, 0, 1, 5], [3, 7, 4, 5, 0, 1, 2, 6], [4, 5, 6, 0, 2, 3, 7, 1], [5, 6, 0, 1, 3, 7, 4, 2], [6, 0, 1, 2, 7, 4, 5, 3], [7, 4, 5, 6, 1, 2, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 2, 7, 5, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 2, 6, 1, 5, 3, 7], [0, 5, 7, 2, 5, 7, 2, 0], [0, 6, 5, 4, 3, 2, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0095","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5e87494df2c6e76585fe0a5909cec18f629ef0d60128dadec10a81fa91d2385","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282095,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 0, 5, 4], [2, 3, 7, 4, 5, 1, 0, 6], [3, 7, 4, 6, 0, 2, 1, 5], [4, 6, 5, 0, 2, 7, 3, 1], [5, 0, 1, 2, 7, 6, 4, 3], [6, 5, 0, 1, 3, 4, 7, 2], [7, 4, 6, 5, 1, 3, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 2, 6, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 2, 5, 1, 3, 6, 7], [0, 5, 6, 4, 3, 1, 2, 7], [0, 6, 7, 2, 6, 2, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0096","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5899ebc2708f7e98ec2717a5eb7ad5b27a68dfb10188a9fa4a194c1cc9b94f1","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282096,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0097","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"252389e2ef50f273841099fca64daac1d045ecc5a7e729b8a4623565f3a2bb15","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282097,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0098","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ac4589f125fb39353cfc572c41305bbcaf88cf06ebe5a13de480d6a16dbbc87","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282098,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 5, 0, 4, 6], [2, 3, 7, 6, 0, 1, 5, 4], [3, 7, 6, 4, 1, 2, 0, 5], [4, 5, 0, 1, 7, 6, 3, 2], [5, 0, 1, 2, 6, 4, 7, 3], [6, 4, 5, 0, 3, 7, 2, 1], [7, 6, 4, 5, 2, 3, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 4, 2, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 2, 4, 0], [0, 5, 4, 6, 2, 1, 3, 7], [0, 6, 2, 5, 4, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0099","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e66fe5fbd36d131c822eb36a1ce903f773ca0b8fb8d0a2b31eb20a73cd5c58e2","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282099,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 4, 5, 6], [2, 3, 7, 6, 1, 0, 4, 5], [3, 7, 6, 5, 2, 1, 0, 4], [4, 0, 1, 2, 5, 6, 7, 3], [5, 4, 0, 1, 6, 7, 3, 2], [6, 5, 4, 0, 7, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 5, 5, 7, 2, 0], [0, 3, 5, 1, 6, 2, 4, 7], [0, 4, 5, 6, 1, 2, 3, 7], [0, 5, 7, 2, 2, 7, 5, 0], [0, 6, 2, 4, 3, 5, 1, 7], [0, 7, 0, 7, 7, 0, 7, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0, 1, 2, 3, 4, 5, 6, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0100","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8af28c8e77c37bc9e253e89e5005a50debc5b13980abce2af091d20397ed400b","family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282100,"semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0]}"},"fields":{"title":"The Zero Ideal and Whole Ring Are Not the Empty Subset","stem":"Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 4, 5, 3, 6, 7, 0], [2, 4, 3, 6, 5, 7, 0, 1], [3, 5, 6, 0, 7, 1, 2, 4], [4, 3, 5, 7, 6, 0, 1, 2], [5, 6, 7, 1, 0, 2, 4, 3], [6, 7, 0, 2, 1, 4, 3, 5], [7, 0, 1, 4, 2, 3, 5, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 3, 0, 6, 2, 3, 6], [0, 3, 0, 0, 3, 3, 0, 3], [0, 4, 6, 3, 1, 7, 2, 5], [0, 5, 2, 3, 7, 1, 6, 4], [0, 6, 3, 0, 2, 6, 3, 2], [0, 7, 6, 3, 5, 4, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Normalize subset=[0] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals).","answer":{"kind":"expression","value":"([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)","explanation":"Compare with the singleton containing the actual additive zero and with the full carrier separately. These two subsets are always two-sided ideals; the supplied subset is still audited rather than assumed to be one of them. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], Yes, No, Yes, zero_and_whole_are_always_two_sided_ideals)."},"skills":["The Zero Ideal and Whole Ring Are Not the Empty Subset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}