CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A module over a Noetherian ring has no associated primes exactly when it is zero

Statement

Assume Dependent Choice.

Let $R$ be a Noetherian commutative ring and let $M$ be a left $R$ -module. Then ${Ass}_{R} (M) = \emptyset ⟺ M = 0.$

Facts & Assumptions

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

[L1]

Under Dependent Choice, every nonzero $R$ -module has an associated prime (A nonzero module over a Noetherian ring has an associated prime).

Proof

technique · direct

1.1givenalgebra

If $M = 0$ , then the only element of $M$ is $0$ , whose annihilator is the whole ring $R$ . Since $R$ is not a prime ideal of itself, no associated prime can occur, so ${Ass}_{R} (M) = \emptyset$ .

1.2L1given

If $M \neq 0$ , then [L1] gives ${Ass}_{R} (M) \neq \emptyset$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the equivalence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)