In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.
The main effect of p having a superfluous kernel is the following: if N is any proper submodule of P, then
[1] Informally speaking, this shows the superfluous kernel causes P to cover M optimally, that is, no submodule of P would suffice.
This does not depend upon the projectivity of P: it is true of all superfluous epimorphisms.
Unlike injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have projective covers.
A ring is called semiperfect if every finitely generated left (right) R-module has a projective cover in R-Mod (Mod-R).
A ring is called lift/rad if idempotents lift from R/J to R, where J is the Jacobson radical of R. The property of being lift/rad can be characterized in terms of projective covers: R is lift/rad if and only if direct summands of the R module R/J (as a right or left module) have projective covers.