LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 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 maximal regular sequence stops at an associated prime

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Let $R$ be Noetherian, let $M$ be finite, let $I$ be an ideal, and let $x = (x_{1}, \dots, x_{r})$ be an $M$ -regular sequence in $I$ . Put $Q = M / (x) M$ and assume $I Q \neq Q$ . Then $x$ is maximal among regular sequences in $I$ if and only if $I \subseteq p$ for some $p \in {Ass}_{R} (Q)$ .

Facts & Assumptions

Given: the Axiom of Choice and the data and terminal quotient in the statement.

[L1]

Under Choice, depth zero of a nonzero finite module with $I Q \neq Q$ is equivalent to $I$ lying in an associated prime (Depth zero and associated primes).

Proof

technique · direct

1.1givenalgebra

The sequence can be extended in $I$ exactly when $I$ contains a nonzerodivisor on $Q$ whose quotient is nonzero. If such an element $y$ had $Q / y Q = 0$ , then $Q = y Q \subseteq I Q$ , contrary to $I Q \neq Q$ .

2.1L1step 1.1∎

Thus maximality is equivalent to ${depth}_{I} (Q) = 0$ . By [L1], this is exactly the displayed containment.

Depends on

Used by

Maximal regular sequences have a common Ext length Lemma

Dependency tree · two levels

8 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)