Question 14626: Keep the Finite Step from Integral Domain to Invertibility

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, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 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, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 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

Step-by-step derivation

same carrier Fin(5);addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]];multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]]

actual additive zero=0;actual multiplicative one=1;additive inverses=[0, 4, 3, 2, 1]

all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null]

additive commute=Yes;multiplicative commute=Yes;zero differs from one=Yes

ordered nonzero labels=[1, 1];actual zero=0;product label=1

nonzero zero-divisor witness=No

ordered nonzero labels=[1, 2];actual zero=0;product label=2

nonzero zero-divisor witness=No

ordered nonzero labels=[1, 3];actual zero=0;product label=3

nonzero zero-divisor witness=No

ordered nonzero labels=[1, 4];actual zero=0;product label=4

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 1];actual zero=0;product label=2

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 2];actual zero=0;product label=4

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 3];actual zero=0;product label=1

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 4];actual zero=0;product label=3

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 1];actual zero=0;product label=3

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 2];actual zero=0;product label=1

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 3];actual zero=0;product label=4

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 4];actual zero=0;product label=2

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 1];actual zero=0;product label=4

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 2];actual zero=0;product label=3

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 3];actual zero=0;product label=2

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 4];actual zero=0;product label=1

nonzero zero-divisor witness=No

all ordered nonzero zero-divisor pairs=[]

integral domain=Yes;invertibility is a later finite theorem step

nonzero element=1;multiplication map on full carrier=[0, 1, 2, 3, 4];n=5

cancellation makes map injective;finite equal-size carrier makes it surjective

actual one=1;preimage candidates in increasing label order=[1]

selected inverse=1;left product=1;right product=1

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, 1, 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.