# Projective cover

Projective cover

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.

Definition

Let $mathcal\left\{C\right\}$ be a category and "X" an object in $mathcal\left\{C\right\}$. A projective cover is a pair ("P","p"), with "P" a projective object in $mathcal\left\{C\right\}$ and "p" a superfluous epimorphism "f" in Hom("P", "X").In the category of "R"-modules, this means that "f(P)" = "X" and $f\left(P\text{'}\right) e X$ for all proper submodules "P' " of "P".

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.

ee also

* Projective resolution
* Injective envelope

References

*cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|date = 1992|isbn = 0387978453|url = http://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27

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