TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A commutative ring is Artinian exactly when it has finite length as a module over itself

Statement

Assume the Axiom of Choice.

Let $R$ be a commutative ring. Then $R$ is Artinian if and only if the regular module $_{R} R$ has finite length.

Facts & Assumptions

Given: A commutative ring $R$ and the Axiom of Choice.

Proof

technique · direct

1.1givenalgebra

If $_{R} R$ has finite length, then by Composition series and length of a module it has a composition series. The forward implication of A module has a composition series if and only if it is Noetherian and Artinian, the converse using dependent choice therefore makes $_{R} R$ Artinian, and Left and right Artinian rings says exactly that $R$ is an Artinian ring.

1.2givenalgebra

Suppose now that $R$ is Artinian. Then Left and right Artinian rings says that the regular module $_{R} R$ is Artinian. Under the Axiom of Choice assumed in the Statement, Every commutative Artinian ring is Noetherian makes $R$ Noetherian, so Left and right Noetherian rings makes $_{R} R$ Noetherian. Since the same choice assumption also suffices for the dependent-choice use recorded in A module has a composition series if and only if it is Noetherian and Artinian, the converse using dependent choice, that theorem gives a composition series for $_{R} R$ .

2.1step 1.1step 1.2givenalgebra

By Composition series and length of a module, a module has finite length exactly when it has a composition series. So step 1.1 proves the forward implication, and step 1.2 proves the reverse implication.

3.1step 2.1∎

Therefore a commutative ring is Artinian exactly when its regular module has finite length.

A commutative ring is Artinian exactly when it has finite length as a module over itself

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources