TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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

Statement

Assume Dependent Choice. 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 the Axiom of Choice also holds and $M$ is finitely generated, this is a finite union.

Facts & Assumptions

Given: Dependent Choice, a Noetherian commutative ring $R$ and a left $R$ -module $M$ . The finite-union clause additionally assumes the Axiom of Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice).

[L1]

Under Dependent Choice, every zero divisor on $M$ lies in an associated prime of $M$ (A zero divisor is contained in an associated prime).

[L2]

Under the additional Axiom of Choice, if $M$ is finitely generated, then ${Ass}_{R} (M)$ is finite (Finite modules over Noetherian rings have finitely many associated primes).

Proof

technique · direct

1.1givenalgebra

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 $M$ .

1.2L1

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

2.1L2step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the union formula under Dependent Choice. Under the additional Axiom of Choice, when $M$ is finitely generated, fact [L2] makes that union finite.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources