Finite two-sided ideal

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

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 — The ideal inherits its operations from the same parent ring.
• Subset — I is an explicit subset of the parent carrier.
• 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.