TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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

1.1

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.

givenalgebra

1.2

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 $_{R} R$ .

givenalgebra

2.1

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.

step 1.1step 1.2givenalgebra

3.1

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

step 2.1∎

Depends on

Used by

A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Corollary (19.15) (standard reference, not scraped)
The Stacks Project, Section 10.53: Artinian rings (standard reference, not scraped)