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| 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 (category theory)|dual]] of [[injective envelope]]s.
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| == Definition ==
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| Let <math>\mathcal{C}</math> be a [[category (mathematics)|category]] and ''X'' 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'', ''X'').
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| If ''R'' is a ring, then in the category of ''R''-modules, a '''superfluous epimorphism''' is then an [[epimorphism]] <math>p : P \to X</math> such that the [[kernel (algebra)|kernel]] of ''p'' is a [[superfluous submodule]] of ''P''.
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| ==Properties==
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| 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]].
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| 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. | |
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| 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 [[epimorphism#Related concepts|split epimorphism]] α from ''P' '' to ''P'' such that <math>p\alpha=p'</math>
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| Unlike [[injective envelope]]s and [[flat cover]]s, which exist for every left (right) [[module (mathematics)|''R''-module]] regardless of the [[ring (mathematics)|ring]] ''R'', left (right) ''R''-modules do not in general have projective covers. A ring ''R'' is called left (right) [[perfect ring|perfect]] if every left (right) ''R''-module has a projective cover in ''R''-Mod (Mod-''R'').
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| A ring is called [[semiperfect ring|semiperfect]] if every [[finitely generated module|finitely generated]] left (right) ''R''-module has a projective cover in ''R''-Mod (Mod-''R''). "Semiperfect" is a left-right symmetric property.
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| A ring is called ''lift/rad'' if [[Idempotent element#Category of R modules|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}}
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| == Examples ==
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| In the category of ''R'' modules:
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| *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.
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| *If J(''R'')=0, then a module ''M'' has a projective cover if and only if ''M'' is already projective.
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| *In the case that a module ''M'' is [[simple module|simple]], then it is necessarily the [[top (mathematics)|top]] of its projective cover, if it exists.
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| *The injective envelope for a module always exists, however over certain rings modules may not have projective covers. The class of rings which provides all of its right modules with projective covers is the class of right [[perfect ring]]s.
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| ==See also==
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| * [[Projective resolution]]
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| == References ==
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| {{reflist}}
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| *{{cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|year = 1992|isbn = 0-387-97845-3|url = http://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27}}
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| *{{Citation|last=Faith|first=Carl|title= Algebra. II. Ring theory.| publisher=Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag|year= 1976}}
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| *{{citation|last= Lam|first=T. Y.|title=A first course in noncommutative rings.|edition= 2nd|publisher=Graduate Texts in Mathematics, 131. Springer-Verlag|year=2001|ISBN=0-387-95183-0}}
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| [[Category:Category theory]]
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| [[Category:Homological algebra]]
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| [[Category:Module theory]]
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