Finite commutative ring

Mathematical statement

Using the same complete addition and multiplication tables of an already qualified finite unital ring, require a*b=b*a for every a,b. This slice checks all element pairs rather than guessing from a few samples or apparent table symmetry.

Representative formula (exact TeX)

C(R)

Assumptions

• The common carrier and both complete tables are explicit and first satisfy all finite unital-ring axioms.
• Commutativity quantifies over every ordered element pair; absence of zero divisors and invertibility of nonzero elements are not additional premises.

Symbol scope

• C — commutative-ring record
• R — parent finite unital ring

Intuition, definition, and misconceptions

Retain the parent ring's zero, one and complete ring axioms.

Check multiplicative commutativity for every pair.

A commutative ring may still have nonzero zero divisors; integral domains and fields are separate nodes.

A few commuting products do not make the whole ring commutative, and a commutative ring is not automatically an integral domain or field.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite unital ring — Commutativity is an additional condition on the same already certified ring tables.
• Quantifiers and explicit domains — Multiplicative commutativity quantifies over all element pairs.

Verification boundary

Only the specified complete finite ring is classified; absence of zero divisors, a field structure or a general infinite-ring conclusion is not inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add finite integral domains and polynomial rings over finite commutative coefficient rings.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.