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

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Zero divisors on a module over a Noetherian ring are the union of its associated primes

Statement

Let $R$ be a Noetherian commutative ring and let $M$ be a left $R$ -module. Then the set of zero divisors on $M$ is

⋃_{p \in {Ass}_{R} (M)} p .

If $M$ is finitely generated, this is a finite union.

Facts & Assumptions

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

[L1]

Every zero divisor on $M$ lies in an associated prime of $M$ (A zero divisor is contained in an associated prime).

[L2]

If $M$ is finitely generated, then ${Ass}_{R} (M)$ is finite (Finite modules over Noetherian rings have finitely many associated primes).

Proof

technique · direct

1.1

If $x \in p$ for some $p \in {Ass}_{R} (M)$ , choose $m \neq 0$ in $M$ with ${Ann}_{R} (m) = p$ . Then $x m = 0$ , so $x$ is a zero divisor on $M$ .

givenalgebra

1.2

Conversely, every zero divisor on $M$ lies in an associated prime by [L1].

2.1

Steps 1.1 and 1.2 prove the union formula. When $M$ is finitely generated, fact [L2] makes that union finite.

L2step 1.1step 1.2∎

Depends on

Used by

Primary submodules of finite modules are characterized by a singleton associated-prime set Theorem

Dependency tree · two levels

6 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

The Stacks Project, Lemma 10.63.9 (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Proposition (17.15) (standard reference, not scraped)