LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Ext degree zero identifies ideal-annihilated elements

Statement

For a commutative ring $R$ , an ideal $I$ , and an $R$ -module $M$ , evaluation at $1 + I$ gives a natural isomorphism ${Ext}_{R}^{0} (R / I, M) = {Hom}_{R} (R / I, M) ≅ (0 :_{M} I) .$

Facts & Assumptions

Given: A ring $R$ , ideal $I$ , and module $M$ .

Proof

technique · direct

1.1givenconstruct

An $R$ -linear map $f : R / I \to M$ is determined by $m = f (1 + I)$ , and $I m = 0$ . Conversely, every $m \in (0 :_{M} I)$ defines $f_{m} (r + I) = r m$ .

2.1step 1.1algebra∎

These constructions are inverse and natural. Since degree-zero Ext is Hom, they give the asserted isomorphism.

Depends on

The balanced Ext bifunctor

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)