# Question 14641: Keep the Finite Step from Integral Domain to Invertibility [Return to catalogue page 74](https://ocme.evemisslab.com/en/questions/page/74/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r01-finite-integral-domain-foundations/#q-20261014-advance-20261029-r01-finite-integral-domain-0091) 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. R0028-14641 · q-20261014-advance-20261029-r01-finite-integral-domain-0091 ## Original question and givens Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective). - G001 — Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective). ## 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-integral-domain](https://ocme.evemisslab.com/en/mko/mko-finite-integral-domain/) ## Step-by-step derivation S001 · impliesBefore same carrier Fin(5);addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]];multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] After actual additive zero=0;actual multiplicative one=3;additive inverses=[0, 3, 4, 1, 2] Conditions: none Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1. Step identifiers and dependenciesRule: commutative_parent_actual_tables_and_identities · Uses: G001 S002 · impliesBefore all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null] After additive commute=Yes;multiplicative commute=Yes;zero differs from one=Yes Conditions: none Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes. Step identifiers and dependenciesRule: complete_commutative_unital_parent_qualification · Uses: G001, S001 S003 · impliesBefore ordered nonzero labels=[1, 1];actual zero=0;product label=3 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S002 S004 · impliesBefore ordered nonzero labels=[1, 2];actual zero=0;product label=4 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S003 S005 · impliesBefore ordered nonzero labels=[1, 3];actual zero=0;product label=1 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S004 S006 · impliesBefore ordered nonzero labels=[1, 4];actual zero=0;product label=2 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S005 S007 · impliesBefore ordered nonzero labels=[2, 1];actual zero=0;product label=4 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S006 S008 · impliesBefore ordered nonzero labels=[2, 2];actual zero=0;product label=1 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S007 S009 · impliesBefore ordered nonzero labels=[2, 3];actual zero=0;product label=2 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S008 S010 · impliesBefore ordered nonzero labels=[2, 4];actual zero=0;product label=3 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S009 S011 · impliesBefore ordered nonzero labels=[3, 1];actual zero=0;product label=1 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S010 S012 · impliesBefore ordered nonzero labels=[3, 2];actual zero=0;product label=2 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S011 S013 · impliesBefore ordered nonzero labels=[3, 3];actual zero=0;product label=3 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S012 S014 · impliesBefore ordered nonzero labels=[3, 4];actual zero=0;product label=4 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S013 S015 · impliesBefore ordered nonzero labels=[4, 1];actual zero=0;product label=2 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S014 S016 · impliesBefore ordered nonzero labels=[4, 2];actual zero=0;product label=3 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S015 S017 · impliesBefore ordered nonzero labels=[4, 3];actual zero=0;product label=4 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S016 S018 · impliesBefore ordered nonzero labels=[4, 4];actual zero=0;product label=1 After nonzero zero-divisor witness=No Conditions: none Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain obligation. Step identifiers and dependenciesRule: domain_all_ordered_nonzero_pairs · Uses: G001, S017 S019 · impliesBefore all ordered nonzero zero-divisor pairs=[] After integral domain=Yes;invertibility is a later finite theorem step Conditions: none Here a domain is a commutative unital parent with no nonzero zero divisors.Nonzero invertibility is not another defining clause;it follows later with a finite-carrier step. Step identifiers and dependenciesRule: domain_definition_not_inverse_definition · Uses: G001, S018 S020 · impliesBefore nonzero element=1;multiplication map on full carrier=[0, 3, 4, 1, 2];n=5 After cancellation makes map injective;finite equal-size carrier makes it surjective Conditions: none Cancellation first gives injectivity.Only then does finiteness of the same source/target carrier give surjectivity.This finite step does not extend to arbitrary infinite domains,such as the integers. Step identifiers and dependenciesRule: finite_injective_to_surjective_indispensable_step · Uses: G001, S019 S021 · impliesBefore actual one=3;preimage candidates in increasing label order=[1] After selected inverse=1;left product=3;right product=3 Conditions: none Surjectivity supplies a preimage of one;choose the least label witness.Commutativity makes the other product one too.Zero is never inverted. Step identifiers and dependenciesRule: finite_domain_identity_preimage_is_inverse · Uses: G001, S020 ## Reverse checks - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (1, 3, 1, Yes, finite_injective_multiplication_implies_surjective) 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: 2548c00095fb78c34a4d7c4443da7153f379c7cdd652064c1cb171ff20776d9b - candidate record SHA-256: cb074ef9f82602637547c5a1aa69e9bcac0a91240106e039c7613b9ed7b6f2e1 - semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae - question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r01-finite-integral-domain-0091.json)