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.

1 Definition
2 Examples
3 See Also
4 References

[edit] 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 in Hom(P, X).

[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.