DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

An essential epimorphism is a surjection with superfluous kernel, and a projective cover is a projective source with such a map

Definition

An essential epimorphism of modules is a surjective homomorphism $π : P \to M$ whose kernel is superfluous: whenever $N \leq P$ and $N + \ker π = P$ , one already has $N = P$ .

A projective cover of $M$ is an essential epimorphism $π : P \to M$ with $P$ projective in the sense of Projective modules and the lifting property and Equivalent characterizations of projective modules.

The superfluous-kernel language is the one used on this page, motivated by For a finite-length module, the radical is a superfluous submodule.

Depends on

Used by

Every finite-dimensional module has a projective cover, unique up to isomorphism over the target Theorem

Dependency tree · two levels

13 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

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)