TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A nonzero module over a Noetherian ring has an associated prime

Statement

Let $R$ be a Noetherian commutative ring and let $M$ be a nonzero left $R$ -module. Then ${Ass}_{R} (M)$ is nonempty.

Facts & Assumptions

Given: A Noetherian commutative ring $R$ and a nonzero left $R$ -module $M$ .

[L1]

A prime ideal belongs to ${Ass}_{R} (M)$ exactly when it is the annihilator of some element of $M$ (Associated primes of a module).

[L2]

Some nonzero element of $M$ has annihilator maximal among annihilators of nonzero elements (A nonzero module over a Noetherian ring has a maximal element annihilator).

[L3]

An annihilator maximal among annihilators of nonzero elements is prime (A maximal element annihilator is prime).

Proof

technique · direct

1.1

By [L2], choose $m \in M$ with $m \neq 0$ such that ${Ann}_{R} (m)$ is maximal among the annihilators of nonzero elements of $M$ . Then [L3] shows that ${Ann}_{R} (m)$ is prime.

L2L3choose

2.1

By [L1], the prime ideal ${Ann}_{R} (m)$ belongs to ${Ass}_{R} (M)$ .

L1step 1.1

3.1

Therefore ${Ass}_{R} (M)$ is nonempty.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

8 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., Proposition (17.13) (standard reference, not scraped)
The Stacks Project, Lemma 10.63.7 (standard reference, not scraped)