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

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Projective cover

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