Quotient ring of a finite ring

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

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 parent ring has both complete operation tables.
• Finite two-sided ideal — The ideal supplies congruence and well-definedness.
• Equivalence relation — Ideal congruence groups quotient elements into classes.
• 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.