Finite integral domain

Mathematical statement

Using the same complete finite commutative unital-ring tables, require a*b to differ from zero for every pair of nonzero elements. The complete table supports an exhaustive zero-divisor audit. Invertibility of every nonzero element is a subsequent theorem using finiteness, not silently inserted into this definition.

Representative formula (exact TeX)

D(R)

Assumptions

• The parent structure already qualifies as a complete finite commutative unital ring.
• The no-zero-divisor obligation quantifies over every pair of nonzero elements; sampled products do not suffice.

Symbol scope

• D — finite integral-domain record
• R — parent commutative ring

Intuition, definition, and misconceptions

The parent unital ring retains 1 distinct from 0.

An exhaustive search for nonzero zero divisors supplies qualification or a counterexample.

In a finite integral domain multiplication by a nonzero element is injective; finiteness then gives surjectivity and an inverse.

Checking a few products does not establish absence of zero divisors, and nonzero invertibility must not be inserted into the definition without the finite-theorem step.

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 commutative ring — The domain uses the same complete commutative-ring tables.
• Proposition — The no-zero-divisor condition can be stated as an implication.
• Quantifiers and explicit domains — The obligation quantifies over every nonzero element pair.

Verification boundary

Only the specified complete finite commutative ring is classified; no arbitrary infinite-ring result or field conclusion that skips the finite step 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

Use finiteness to prove every nonzero element invertible, then add finite fields.

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.