LemmaStatement: 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.

A prime minimal over an associated prime plus one element becomes associated after a power quotient

Statement

Assume the Axiom of Choice. Let $R$ be Noetherian, $M$ finite, $p \in {Ass}_{R} (M)$ , and $x \in R$ . If $q$ is minimal over $p + (x)$ , then for some $n \geq 1$ , $q \in {Ass}_{R} (M / x^{n} M)$ .

Facts & Assumptions

Given: The Axiom of Choice and the data in the statement.

[L1]

Under AC, every prime minimal in the support of a finite module over a Noetherian ring is associated (Minimal support primes of a finite module are associated).

Proof

technique · direct

1.1givenconstruct

Embed $N = R / p$ in $M$ . Artin--Rees gives $n \geq 1$ such that $N \cap x^{n} M \subseteq x N$ . Hence $L = N / (N \cap x^{n} M) \subseteq M / x^{n} M$ has quotient $N / x N ≅ R / (p + (x))$ .

2.1L1step 1.1given∎

We have $x^{n} N \subseteq N \cap x^{n} M \subseteq x N$ . Viewing $N$ as $R / p$ , the annihilator of $L$ therefore lies between $p + (x^{n})$ and $p + (x)$ , so its radical is $\sqrt{p + (x)}$ . Thus $q$ is minimal in $Supp (L)$ and, under the stated AC, associated to $L$ by [L1]. An associated element of the submodule $L$ has the same annihilator in $M / x^{n} M$ .

Depends on

Used by

Depth is bounded by the quotient dimension at every associated prime Lemma

Dependency tree · two levels

14 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)