CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Depth zero and associated primes

Statement

Assume the Axiom of Choice.

Let $R$ be Noetherian, let $M \neq 0$ be a finite $R$ -module, and let $I$ be an ideal with $I M \neq M$ . Then ${depth}_{I} (M) = 0 ⟺ I \subseteq p for some p \in {Ass}_{R} (M) .$

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice); the ring, module, and ideal in the statement.

Proof

technique · direct

1.1given

Because $I M \neq M$ , depth zero means precisely that the empty regular sequence cannot be extended by an $M$ -regular element of $I$ .

2.1step 1.1∎

The regular-element lemma identifies failure of such an extension with $I$ being contained in an associated prime of $M$ . This proves the biconditional.

Depends on

Used by

Dependency tree · two levels

9 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)