ExampleConstruction: Literature-sourcedVerification: 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.

The zero-module and surjective-ideal depth conventions

Example

For every commutative ring $R$ and ideal $I$ , ${depth}_{I} (0) = + \infty$ . There are also nonzero examples with $I M = M$ : for $R = k \times k$ , $I = k \times 0$ , and $M = I$ , one has $I M = M$ and hence ${depth}_{I} (M) = + \infty$ .

Facts & Assumptions

Given: $M = I$ is generated by the idempotent $(1, 0)$ and is nonzero.

Verification

technique · direct

1.1

The exceptional clause in the definition applies to the zero module because $I 0 = 0$ . In the product-ring example, $I^{2} = I$ , so $I M = I^{2} = M$ .

given

2.1

lem-depth-infinity-when-ideal-acts-surjectively therefore assigns $+ \infty$ in both cases. This does not conflict with Nakayama: the ideal in the nonzero example is not contained in the Jacobson radical.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)