# Question 13718: Check the Additive Structure before Multiplicative Absorption [Return to catalogue page 69](https://ocme.evemisslab.com/en/questions/page/69/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations/#q-20261014-advance-20261028-r02-finite-two-sided-ideal-0043) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0026-13718 · q-20261014-advance-20261028-r02-finite-two-sided-ideal-0043 ## Original question and givens 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?). - G001 — 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?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-finite-two-sided-ideal](https://ocme.evemisslab.com/en/mko/mko-finite-two-sided-ideal/) ## Step-by-step derivation S001 · impliesBefore same parent carrier Fin(8);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]] After both complete operation tables typed and closed Conditions: none Bind subsequent ideals or commutativity to this same parent ring.Preserve the complete original tables;carrier labels are not integer arithmetic values. Step identifiers and dependenciesRule: parent_ring_original_complete_tables · Uses: G001 S002 · impliesBefore every additive triple checked;first associativity failure=null;all additive pairs commute=Yes After additive zero candidates=[0];actual zero=0;two-sided additive inverses in label order=[0, 4, 5, 7, 1, 2, 6, 3] Conditions: none Audit additive associativity,commutativity,the two-sided zero and each inverse over the complete carrier.This qualified parent is the problem's premise,checked by complete finite conditions rather than samples. Step identifiers and dependenciesRule: parent_additive_abelian_group_complete_audit · Uses: G001, S001 S003 · impliesBefore every multiplicative triple checked;first associativity failure=null After multiplicative identity candidates=[1];actual one=1 Conditions: none Multiplication here requires associativity and a two-sided one,not commutativity or inverses of every nonzero element. Step identifiers and dependenciesRule: parent_multiplicative_monoid_complete_audit · Uses: G001, S002 S004 · impliesBefore all 512 left and 512 right distributivity triples checked After first left failure=null;first right failure=null;zero=0;one=1;distinct=Yes Conditions: none Audit both distributive obligations completely and separately.Zero and one must be distinct actual elements.Unital alone implies neither a commutative ring,domain nor field. Step identifiers and dependenciesRule: parent_both_distributivities_and_distinct_identities · Uses: G001, S003 S005 · impliesBefore raw subset=[0];same qualified parent zero=0 After sorted distinct parent labels=[0] Conditions: none Normalize the subset as a sorted set,but obtain all operations and inverses from the same parent ring.Do not substitute new operations or integer arithmetic on labels. Step identifiers and dependenciesRule: ideal_subset_same_parent_normalization · Uses: G001, S004 S006 · impliesBefore subset element=0;parent additive inverse=0 After inverse belongs to subset=Yes Conditions: none An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse. Step identifiers and dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S005 S007 · impliesBefore subset ordered pair=[0, 0];parent sum=0 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S006 S008 · impliesBefore subset nonempty=Yes;actual zero belongs=Yes;missing sums=[];missing inverses=[] After additive subgroup=Yes Conditions: none Nonemptiness,the actual zero,addition closure and inverse closure must all hold.An empty subset does not become an additive subgroup through vacuous closure checks. Step identifiers and dependenciesRule: ideal_additive_subgroup_nonempty_zero_and_closure · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0], 0, [], [], Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 66ca8f9db9b8eb5065ec8575fdf552c65d54ac46a757200395b487e04aa295ee - candidate record SHA-256: 9cc623bda51044b923ed23fa1602779e97d9ec0f586cc0f024707216e38d5b85 - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r02-finite-two-sided-ideal-0043.json)