In the branch of abstract mathematics calledcategory theory, aprojective cover of an objectX is in a sense the best approximation ofX by aprojective objectP. Projective covers are thedual ofinjective envelopes.

Let${\displaystyle \mathcal{C}}$ be acategory andX an object in${\displaystyle \mathcal{C}}$. Aprojective cover is a pair (P,p), withP aprojective object in${\displaystyle \mathcal{C}}$ andp a superfluous epimorphism in Hom(P,X).

IfR is a ring, then in the category ofR-modules, asuperfluous epimorphism is then anepimorphism${\displaystyle p:P\to X}$ such that thekernel ofp is asuperfluous submodule ofP.

Projective covers and their superfluous epimorphisms, when they exist, are unique up toisomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledgeduniversal property.

The main effect ofp having a superfluous kernel is the following: ifN is any proper submodule ofP, then${\displaystyle p(N)\ne M}$.^[1] Informally speaking, this shows the superfluous kernel causesP to coverM optimally, that is, no submodule ofP would suffice. This does not depend upon the projectivity ofP: it is true of all superfluous epimorphisms.

If (P,p) is a projective cover ofM, andP' is another projective module with an epimorphism${\displaystyle p^{\prime }:P^{\prime }\to M}$, then there is asplit epimorphism α fromP' toP such that${\displaystyle p\alpha =p^{\prime }}$

Unlikeinjective envelopes andflat covers, which exist for every left (right)R-module regardless of theringR, left (right)R-modules do not in general have projective covers. A ringR is called left (right)perfect if every left (right)R-module has a projective cover inR-Mod (Mod-R).

A ring is calledsemiperfect if everyfinitely generated left (right)R-module has a projective cover inR-Mod (Mod-R). "Semiperfect" is a left-right symmetric property.

A ring is calledlift/rad ifidempotents lift fromR/J toR, whereJ is theJacobson radical ofR. 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 theR moduleR/J (as a right or left module) have projective covers.^[2]

In the category ofR modules:

• IfM is already a projective module, then the identity map fromM toM is a superfluous epimorphism (its kernel being zero). Hence, projective modules always have projective covers.
• If J(R)=0, then a moduleM has a projective cover if and only ifM is already projective.
• In the case that a moduleM issimple, then it is necessarily thetop of its projective cover, if it exists.
• The injective envelope for a module always exists, however over certain rings modules may not have projective covers. For example, the natural map fromZ ontoZ/2Z is not a projective cover of theZ-moduleZ/2Z (which in fact has no projective cover). The class of rings which provides all of its right modules with projective covers is the class of rightperfect rings.
• AnyR-moduleM has aflat cover, which is equal to the projective cover ifM has a projective cover.

• Projective resolution

• Anderson, Frank Wylie; Fuller, Kent R (1992).Rings and Categories of Modules. Springer.ISBN 0-387-97845-3. Retrieved2007-03-27.
• Faith, Carl (1976),Algebra. II. Ring theory., Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag
• Lam, T. Y. (2001),A first course in noncommutative rings (2nd ed.), Graduate Texts in Mathematics, 131. Springer-Verlag,ISBN 0-387-95183-0

• Category theory
• Homological algebra
• Module theory