# Finite commutative ring

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist.

mko-finite-commutative-ring · definition · v0.10.0



## 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

[Finite commutative ring: checking commutativity of ℤ₆ over all element pairs, with an upper-triangular matrix ring as a non-commutative counterexample](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-commutative-ring-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-unital-ring/) — Commutativity is an additional condition on the same already certified ring tables.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-commutative-ring.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-commutative-ring/index.md)