TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-26

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.

The local depth-zero associated-prime criterion

Statement

Assume the Axiom of Choice. Let $(R, m)$ be a Noetherian local ring and let $M \neq 0$ be a finite $R$ -module. Then $depth (M) = 0 ⟺ m \in {Ass}_{R} (M) .$

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian local ring and a nonzero finite module.

[F1]

Assuming the Axiom of Choice, Nakayama's lemma: Under the assumed Choice, if $I$ lies in the Jacobson radical and $M$ is finitely generated, then $I M = M$ implies $M = 0$ .

[F2]

Depth zero and associated primes: Under the assumed Choice, if $I M \neq M$ for a nonzero finite module over a Noetherian ring, then its depth with respect to $I$ is zero exactly when $I$ is contained in an associated prime.

Proof

technique · direct

1.1givenF1F2

Since $M \neq 0$ , [F1] gives $m M \neq M$ , so [F2] applies with $I = m$ . Its published proof uses finiteness of associated primes through an AC-qualified prime filtration; the assumed Choice supplies that premise. It says depth is zero exactly when $m \subseteq p$ for some associated prime $p$ .

2.1step 1.1algebra∎

Every associated prime is proper and every proper ideal of a local ring is contained in $m$ . Hence $m \subseteq p$ forces $p = m$ , proving both directions.

Depends on

Dependency tree · 2 levels

Sources

Depth and Cohen--Macaulay modules source treatment