# Finite integral domain

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

A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors.

mko-finite-integral-domain · definition · v0.10.0



## 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

[Finite integral domain: deciding ℤ/5ℤ and ℤ/6ℤ from complete tables, then deriving invertibility from finiteness](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-integral-domain-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-commutative-ring/) — The domain uses the same complete commutative-ring tables.
- [Proposition](https://ocme.evemisslab.com/en/mko/mko-proposition/) — The no-zero-divisor condition can be stated as an implication.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-integral-domain.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-integral-domain/index.md)