# Finite two-sided ideal

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

A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides.

mko-finite-two-sided-ideal · definition · v0.10.0



## Mathematical statement

Given a finite unital ring R and a fully specified subset I, require I to be nonempty and closed under addition and additive inverses. For every r in R and x in I, require both r*x and x*r to belong to I. Qualification always refers to the same parent ring.





## Representative formula (exact TeX)

I(R)





## Assumptions


- The parent ring qualifies as a finite unital ring using both complete tables, and I is fully specified by distinct carrier elements.
- Left and right absorption are quantified separately; finite samples do not prove two-sidedness.





## Symbol scope


- I — two-sided ideal
- R — parent ring





## Intuition, definition, and misconceptions

Additive-subgroup qualification includes zero and additive inverses.

Left and right absorption each check every parent-ring element.

The zero ideal and whole ring are both two-sided ideals; subrings and one-sided ideals must not be conflated with them.

Multiplicative closure alone, left absorption alone, or a subset without a fixed parent ring does not establish a two-sided ideal.





## Adopted derivation guide

[Finite two-sided ideal: checking left and right absorption separately in an upper-triangular matrix ring](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-two-sided-ideal-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 ideal inherits its operations from the same parent ring.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — I is an explicit subset of the parent carrier.
- [Finite abelian group](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/) — I must form a subgroup under addition.





## Verification boundary

Only the specified complete finite parent ring and subset are checked; no quotient ring, maximal ideal or prime ideal is asserted.

not_formalized

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





## Next obligation

Add quotient rings of finite rings and well-definedness under representative choice.





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