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

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A nonzero module over a Noetherian ring has a maximal element annihilator

Statement

Assume Dependent Choice.

Let $R$ be a Noetherian commutative ring and let $M$ be a nonzero left $R$ -module. Then there exists a nonzero element $m \in M$ such that ${Ann}_{R} (m)$ is maximal, under inclusion, among the annihilators of nonzero elements of $M$ .

Facts & Assumptions

Given: Dependent Choice, a Noetherian commutative ring $R$ , and a nonzero left $R$ -module $M$ .

[L1]

Assuming Dependent Choice, in a Noetherian commutative ring every nonempty set of ideals has a maximal member (A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member).

[L2]

For $m \in M$ , the annihilator is the set ${Ann}_{R} (m) = {r \in R : r m = 0}$ (Annihilators, torsion elements and the torsion subset of a module).

Proof

technique · direct

1.1L2givenchoosealgebra

For every $m \in M$ , the set ${Ann}_{R} (m)$ is an ideal: it contains $0$ , is closed under subtraction, and $a \in {Ann}_{R} (m)$ implies $r a \in {Ann}_{R} (m)$ for every $r \in R$ . Because $M \neq 0$ , choose $m_{0} \in M$ with $m_{0} \neq 0$ . Then $Σ = {{Ann}_{R} (m) : m \in M, m \neq 0}$ is a nonempty set of ideals of $R$ .

2.1L1step 1.1

By [L1], the nonempty set $Σ$ has a maximal member. Thus some nonzero $m \in M$ satisfies that ${Ann}_{R} (m)$ is maximal among the annihilators of nonzero elements of $M$ .

3.1step 2.1∎

This is exactly the required conclusion.

Depends on

Used by

A nonzero module over a Noetherian ring has an associated prime Theorem

Dependency tree · two levels

18 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., Lemma (17.12) (standard reference, not scraped)
The Stacks Project, Section 10.63: Associated primes (standard reference, not scraped)