Projective cover

Summary

A projective cover of a module $M$ is a surjection from a projective module onto $M$ that is as small as possible. This means that its kernel is a superfluous submodule. When it exists it is unique up to isomorphism. It is the dual notion of the injective hull, which always exists, whereas projective covers exist only over special rings. The rings over which every finitely generated module has a projective cover are the semiperfect rings. Those over which every left module has one are the left perfect rings.