Alphabeta Math
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.

Depth is infinite when the ideal acts surjectively

Statement

Let R be a commutative ring, I⊆R an ideal, and M a finite R-module. If IM=M, then depth⁡I(M)=∞. If moreover the Axiom of Choice holds, (R,m) is local, I⊆m, and M≠0, then IM≠M.

Facts & Assumptions

Given: The ring, ideal, and finite module in the statement; AC for the second assertion.

Proof

technique · direct
1.1given

The first assertion is exactly the exceptional convention in the definition of I-depth; it includes M=0.

2.1step 1.1algebra∎

In the local case, IM=M with I⊆m would give mM=M. Under the stated AC hypothesis, Assuming the Axiom of Choice, Nakayama's lemma then forces M=0, contrary to the hypothesis.

Depends on

Used by

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