# Finite unital ring

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

On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero.

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



## Mathematical statement

Supply complete addition and multiplication tables on the same finite carrier X. Addition must form an abelian group; multiplication must be associative and distribute over addition on both sides. There must be a unique multiplicative identity 1, distinct from the additive identity 0. This slice does not require multiplication to commute.





## Representative formula (exact TeX)

R(X)





## Assumptions


- Carrier labels run from 0 to n−1 and both n-by-n operation tables are complete.
- Qualification checks every relevant element pair or triple exactly; finite samples do not replace the ring axioms.





## Symbol scope


- R — finite unital-ring record
- X — common finite carrier
- 0 — additive identity
- 1 — multiplicative identity





## Intuition, definition, and misconceptions

First find zero and additive inverses from the addition table, then check multiplicative associativity and both distributive laws.

The multiplicative identity must act on both sides and be different from zero.

Commutative rings, integral domains, fields, ideals and quotient rings are separate nodes, not consequences of this qualification.

A multiplicative identity alone, a few passing distributivity samples, or multiplicative commutativity is not the whole definition in this unital-ring protocol.





## Adopted derivation guide

[Finite unital ring: proving that the mod-4 addition and multiplication tables form a unital ring](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-unital-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 abelian group](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/) — The same addition table must supply the ring's additive abelian-group structure.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Addition and multiplication are binary operations on one common carrier.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The two complete tables represent total functions from X×X to X.





## Verification boundary

Only the supplied complete finite pair of tables is classified; no general-ring, commutativity, absence-of-zero-divisors or invertibility-of-every-nonzero-element claim follows.

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 commutative rings, two-sided ideals and quotient 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-unital-ring.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-unital-ring/index.md)