# Projective cover

In the branch of abstract mathematics called 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 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 in Hom(*P*, *X*).

If *R* is a ring, then in the category of *R*-modules, a **superfluous epimorphism** is then an epimorphism such that the kernel of *p* is a superfluous submodule of *P*.

## Properties

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

The main effect of *p* having a superfluous kernel is the following: if *N* is any proper submodule of *P*, then .[1] Informally speaking, this shows the superfluous kernel causes *P* to cover *M* optimally, that is, no submodule of *P* would suffice. This does not depend upon the projectivity of *P*: it is true of all superfluous epimorphisms.

If (*P*,*p*) is a projective cover of *M*, and *P' * is another projective module with an epimorphism , then there is a split epimorphism α from *P' * to *P* such that

Unlike injective envelopes and flat covers, 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.

A ring is called *lift/rad* if idempotents lift from *R*/*J* to *R*, where *J* is the Jacobson radical of *R*. 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 the *R* module *R*/*J* (as a right or left module) have projective covers.[2]

## Examples

In the category of *R* modules:

- If
*M*is already a projective module, then the identity map from*M*to*M*is a superfluous epimorphism (its kernel being zero). Hence, projective modules always have projective covers. - If J(
*R*)=0, then a module*M*has a projective cover if and only if*M*is already projective. - In the case that a module
*M*is simple, then it is necessarily the top 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 from
**Z**onto**Z**/2**Z**is not a projective cover of the**Z**-module**Z**/2**Z**(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 right perfect rings. - Any
*R*-module*M*has a flat cover, which is equal to the projective cover if*R*has a projective cover.

## See also

## References

- Proof: Let
*N*be proper in*P*and suppose*p*(*N*)=*M*. Since ker(*p*) is superfluous, ker(*p*)+*N*≠*P*. Choose*x*in*P*outside of ker(*p*)+*N*. By the surjectivity of*p*, there exists*x'*in*N*such that*p*(*x'*)=*p*(*x*),, whence*x*−*x'*is in ker(*p*). But then*x*is in ker(*p*)+*N*, a contradiction. - Anderson & Fuller 1992, p. 302.

- Anderson, Frank Wylie; Fuller, Kent R (1992).
*Rings and Categories of Modules*. Springer. ISBN 0-387-97845-3. Retrieved 2007-03-27. - 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