{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0070", "pilot_number": "R0030-15970", "number": 15970, "family_id": "qf-mod31-quotring-multiply", "title": { "zh-Hant": "第 15970 題：有限商環", "en": "Question 15970: Quotient Multiplication Must Respect the Same Ideal" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0070", "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-quotient-ring" ], "title_zh": "有限商環", "stem_zh": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":30,\"d\":6,\"a\":19,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。", "answer": { "kind": "expression", "value": "(1, 3, 21, 3, [a][b]=[ab])", "explanation_zh": "Wave4收束核對=(1, 3, 21, 3, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限商環" ] }, "variant": { "family_id": "qf-mod31-quotring-multiply", "seed": 202610314070, "parameters": { "n": 30, "d": 6, "a": 19, "b": 9, "domain": "Exact_quotient_of_Zn_by_proper_ideal", "operation": "quotring-multiply", "semantic_template": "qf-mod31-quotring-multiply/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0070", "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e684506a9b4733b2fc66f77f451eab8d382974e2fc14d96e49edf1ded500e478", "family_id": "qf-mod31-quotring-multiply", "seed": 202610314070, "semantic_template": "qf-mod31-quotring-multiply/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}" }, "fields": { "title": "Quotient Multiplication Must Respect the Same Ideal", "stem": "Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).", "answer": { "kind": "expression", "value": "(1, 3, 21, 3, [a][b]=[ab])", "explanation": "Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 21, 3, [a][b]=[ab])." }, "skills": [ "Quotient Multiplication Must Respect the Same Ideal" ], "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." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":30,\"d\":6,\"a\":19,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。", "en": "Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab])." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "quotient ring parent modulus divided by ideal generator := div(30, 6)", "after": "quotient ring parent modulus divided by ideal generator = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 quotient ring parent modulus divided by ideal generator，不以小數近似替代。", "en": "Recompute quotient ring parent modulus divided by ideal generator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "30", "6" ], "output": "5", "label": "quotient ring parent modulus divided by ideal generator" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "same declared parent Z_30;generator d=6;n/d=5;ideal multiples=[0, 6, 12, 18, 24]", "after": "I=kernel of residue map Z_30 to Z_6;proper two-sided ideal=是;zero ideal allowed when d=n", "relation": "implies", "rule_id": "quotient_ring_actual_parent_proper_two_sided_ideal", "conditions": [], "explanation": { "zh-Hant": "因d整除n，整數代表改加n仍不改mod d，π:Z_n→Z_d良定且保加乘。核I由d倍數組成，加法差封閉且左右乘任意母環元素仍在I。d≥2使1不在I，故proper；d=n可得零理想，不把它排除。", "en": "Since d divides n,changing an integer representative by n does not change mod d.The map π:Z_n→Z_d is well-defined and preserves addition and multiplication.Its kernel consists of multiples of d,closed under differences and multiplication on either side by parent elements.With d≥2 it excludes1 and is proper;d=n gives the allowed zero ideal." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29];same ideal I=[0, 6, 12, 18, 24]", "after": "all additive cosets in representative order0..d-1=[[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]];quotient size=6;no field conclusion", "relation": "implies", "rule_id": "quotient_ring_full_cosets_not_representative_carrier", "conditions": [], "explanation": { "zh-Hant": "每個加法coset列全體同餘mod d的母環元素，按代表0..d−1排序；它們互斥且覆蓋Z_n。商元素是整類，輸出class標籤只是索引，不因有限商環就宣稱域。", "en": "List every parent element congruent mod d in each additive coset,ordered by representatives0 through d−1.They are disjoint and cover Z_n.Quotient elements are whole cosets;class labels are indices only,and a finite quotient ring is not automatically a field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "first input coset exact quotient fraction := div(19, 6)", "after": "first input coset exact quotient fraction = 19/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 first input coset exact quotient fraction，不以小數近似替代。", "en": "Recompute first input coset exact quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "19", "6" ], "output": "19/6", "label": "first input coset exact quotient fraction" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "first input coset:integer value=19;positive modulus=6;exact ratio=19/6", "after": "floor quotient=3;quotient<=ratio