In the branch of abstract mathematics called category theory, a '''projective cover''' of an object ''M'' is in a sense the best approximation of ''M'' by a projective object ''P''. Projective covers are the dual of injective envelopes.
== Definition ==
Let <math>\mathcal{C}</math> be a category and ''M'' an object in <math>\mathcal{C}</math>. A '''projective cover''' is a pair (''P'',''p''), with ''P'' a projective object in <math>\mathcal{C}</math> and ''p'' a superfluous epimorphism in Hom(''P'', ''M'').
If ''R'' is a ring, then in the category of ''R''-modules, a '''superfluous epimorphism''' is then an epimorphism <math>p : P \to M</math> 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 <math>p(N) \ne M</math>.<ref>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.</ref> 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 <math>p':P'\rightarrow M</math>, then there is a split epimorphism α from ''P' '' to ''P'' such that <math>p\alpha=p'</math>
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.{{sfn|Anderson|Fuller|1992|loc=p. 302}}
== 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 ''M'' has a projective cover.
==See also== * Projective resolution
== References == {{reflist}} *{{cite book|last = Anderson|first = Frank Wylie|last2=Fuller |first2=Kent R |title = Rings and Categories of Modules|publisher = Springer|year = 1992|isbn = 0-387-97845-3|url = https://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27}} *{{Citation|last=Faith|first=Carl|title= Algebra. II. Ring theory.| publisher=Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag|year= 1976}} *{{citation|last= Lam|first=T. Y.|author-link=Tsit Yuen Lam|title=A first course in noncommutative rings|edition= 2nd|publisher=Graduate Texts in Mathematics, 131. Springer-Verlag|year=2001|isbn=0-387-95183-0}}
Category:Category theory Category:Homological algebra Category:Module theory