Projective cover
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In 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.
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[edit] Definition
Let 
be a category and X an object in . A projective cover is a pair (P,p), with P a projective object in and p a superfluous epimorphism f in Hom(P, X). In the category of R-modules, this means that f(P) = X and 
for all proper submodules P' of P.
[edit] Examples
- R-Mod (Mod-R)
Unlike injective envelopes, 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.
[edit] See also
[edit] References
- Anderson, Frank Wylie; Fuller, Kent R (1992). Rings and Categories of Modules. Springer. ISBN 0387978453. Retrieved on 2007-03-27.
