DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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 modules and the lifting property

Definition

A left $R$ -module $P$ is projective if it has the lifting property for epimorphisms: whenever $q : E \to M$ is a surjective module homomorphism and $f : P \to M$ is a module homomorphism, there exists a module homomorphism $\tilde{f} : P \to E$ such that $q \circ \tilde{f} = f$ (Module homomorphism and isomorphism, kernel, image and cokernel, Injection, surjection, bijection).

The lift need not be unique. Projectivity asks for a lift in every such square.

Projective modules and the lifting property

Definition

Depends on

Dependency tree · 2 levels

Sources