Question 14671: An Integral Domain Requires Every Nonzero Pair to Avoid Zero

Original question and givens

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] 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 (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.

• G001 — Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] 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 (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.

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(4);addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]];multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]]

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

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=3;product label=3

nonzero zero-divisor witness=Yes

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

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

nonzero zero-divisor witness=No

all ordered nonzero zero-divisor pairs=[[0, 0]]

integral domain=No;invertibility is a later finite theorem step

Reverse checks

• 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

(4, 3, 1, [[0, 0]], No, commutative_ring_with_zero_divisors)

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.