LemmaStatement: 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.

Projective left and right modules are flat over an arbitrary ring

Statement

Every projective left or right module over an arbitrary ring is flat on its appropriate side.

Proof

Given: a projective left $R$ -module $P$ ; the right-module case is symmetric.

1.1

Choose a free module $F$ and a module $P^{'}$ with $F ≅ P \oplus P^{'}$ .

given

2.1

For every exact sequence of right modules, tensoring with $F$ is a direct sum of copies of that sequence and is exact; tensoring with $P$ is a direct summand of this exact complex.

step 1.1algebra

3.1

A direct summand of an exact complex is exact, so $- \otimes_{R} P$ is exact and $P$ is flat. The same direct-summand argument proves the right-handed statement.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

16 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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)