Projective cover

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.
Definition.
Let formula_1 be a category and "X" an object in formula_1. A projective cover is a pair ("P","p"), with "P" a projective object in formula_1 and "p" a superfluous epimorphism in Hom("P", "X").
If "R" is a ring, then in the category of "R"-modules, a superfluous epimorphism is then an epimorphism formula_4 such that the kernel of "p" is a superfluous submodule of "P".
Properties.
Projective covers and their superfluous epimorphisms, when they exist, are unique up to isomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledged universal property.
The main effect of "p" having a superfluous kernel is the following: if "N" is any proper submodule of "P", then formula_5. 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.
If ("P","p") is a projective cover of "M", and "P' " is another projective module with an epimorphism formula_6, then there is a split epimorphism α from "P' " to "P" such that formula_7
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 "R" is called left (right) perfect if every left (right) "R"-module has a projective cover in "R"-Mod (Mod-"R").
A ring is called semiperfect if every finitely generated left (right) "R"-module has a projective cover in "R"-Mod (Mod-"R"). "Semiperfect" is a left-right symmetric property.
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.
Examples.
In the category of "R" modules: