Question 14700: Audit All Nonzero Zero-Divisor Pairs

Original question and givens

Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] 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. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.

• G001 — Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] 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. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.

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

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

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=[0, 0];actual zero=1;product label=0

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 2];actual zero=1;product label=4

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

ordered nonzero labels=[2, 4];actual zero=1;product label=3

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 3];actual zero=1;product label=4

nonzero zero-divisor witness=No

ordered nonzero labels=[3, 4];actual zero=1;product label=2

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 2];actual zero=1;product label=3

nonzero zero-divisor witness=No

ordered nonzero labels=[4, 3];actual zero=1;product label=2

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

all ordered nonzero zero-divisor pairs=[]

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

Reverse checks

• 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

([], 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.