Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Hereditary rings have global dimension at most one

Statement

If every submodule of a projective left R-module is projective, then the left global dimension of R is at most one. The right-handed statement holds with right modules throughout.

Facts & Assumptions

Given: The stated left hereditary hypothesis and a left R-module M.

Proof

technique · direct
1.1

Choose a surjection P→M from a free, hence projective, left module P. Its kernel K is a submodule of P and is projective by hypothesis.

givenconstruct
2.1

Thus 0→K→P→M→0 is a projective resolution of length at most one. The definitions Projective dimension of an object and Left and right global dimension of a ring yield the claimed bound; the right proof is identical with sides reversed.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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