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 p(N) \ne M. 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 p':P'\rightarrow M, then there is a
split epimorphism α from ''P'
to P'' such that p\alpha=p' 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 ==