# Quotient ring of a finite ring

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

Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice.

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



## Mathematical statement

Define a equivalent to b when a−b belongs to I. The classes are additive cosets, with [a]+[b]=[a+b] and [a][b]=[ab]. Check representative independence completely. Because this project requires 1 distinct from 0 in a unital ring, the degenerate zero ring from I=R is excluded.





## Representative formula (exact TeX)

Q(R,I)





## Assumptions


- The complete finite parent ring and a certified proper two-sided ideal are explicit.
- All cosets are normalized, and operations are checked for well-definedness across representatives.





## Symbol scope


- Q — finite quotient ring
- I — proper two-sided ideal





## Intuition, definition, and misconceptions

The carrier is formed from additive cosets.

Check that addition and multiplication give the same classes across representative choices.

I=R is separately marked as outside the 1 distinct from 0 convention.

Chosen representatives are not the cosets themselves, and quotient multiplication cannot be declared before proving it well-defined.





## Adopted derivation guide

[Quotienting Z/4Z by its even ideal](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-quotient-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/) — The parent ring has both complete operation tables.
- [Finite two-sided ideal](https://ocme.evemisslab.com/en/mko/mko-finite-two-sided-ideal/) — The ideal supplies congruence and well-definedness.
- [Equivalence relation](https://ocme.evemisslab.com/en/mko/mko-equivalence-relation/) — Ideal congruence groups quotient elements into classes.
- [Quotient set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) — The carrier consists of distinct full classes.





## Verification boundary

Only proper ideals are handled; the quotient is not asserted to be a field or integral domain.

not_formalized

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





## Next obligation

Add prime and maximal ideals.





## 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-quotient-ring.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-ring/index.md)