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CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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The prime ideals of an Artinian ring are exactly its finitely many maximal ideals

Statement

Let R be a commutative Artinian ring. Then the prime ideals of R are exactly its maximal ideals, and this set is finite.

Facts & Assumptions

Given: A commutative Artinian ring R.

Proof

technique · direct
1.1

By Every prime ideal of an Artinian ring is maximal, every prime ideal of R is maximal. By An Artinian ring has only finitely many maximal ideals, the set of maximal ideals is finite.

givenalgebra
2.1

Conversely every maximal ideal of a commutative ring is prime by Every maximal ideal of a commutative ring is prime. Therefore the prime ideals of R are exactly its maximal ideals, and there are only finitely many of them.

step 1.1givenalgebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources