DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 with respect to an ideal

Definition

Let $R$ be a commutative ring, $I \subseteq R$ an ideal, and $M$ a finite $R$ -module. If $I M \neq M$ , define ${depth}_{I} (M) = \sup {r \in N : there is an M -regular sequence of length r in I} .$ If $I M = M$ , set ${depth}_{I} (M) = \infty$ . In particular, the zero module has depth $\infty$ . For a local ring $(R, m)$ , write $depth (M)$ for ${depth}_{m} (M)$ .

Depth with respect to an ideal

Definition

Depends on

Dependency tree · 2 levels

Sources