Projective cover

Projective cover

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.


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 "f" in Hom("P", "X").In the category of "R"-modules, this means that "f(P)" = "X" and f(P') e X for all proper submodules "P' " of "P".


* "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.

ee also

* Projective resolution
* Injective envelope


*cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|date = 1992|isbn = 0387978453|url =|accessdate = 2007-03-27

Wikimedia Foundation. 2010.

См. также в других словарях:

  • Projective unitary group — In mathematics, the projective unitary group PU( n ) is the quotient of the unitary group U( n ) by the right multiplication of its center, U(1), embedded as scalars.Abstractly, it is the isometry group of complex projective space, just as the… …   Wikipedia

  • Projective line — In mathematics, a projective line is a one dimensional projective space. The projective line over a field K , denoted P1( K ), may be defined as the set of one dimensional subspaces of the two dimensional vector space K 2 (it does carry other… …   Wikipedia

  • Real projective space — In mathematics, real projective space, or RP n is the projective space of lines in R n +1. It is a compact, smooth manifold of dimension n , and a special case of a Grassmannian.ConstructionAs with all projective spaces, RP n is formed by taking… …   Wikipedia

  • Fake projective plane — For Freedman s example of a non smoothable manifold with the same homotopy type as the complex projective plane, see 4 manifold. In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have… …   Wikipedia

  • Cycle double cover — Unsolved problems in mathematics Does every bridgeless graph have a multiset of cycles covering every edge exactly twice? …   Wikipedia

  • Flat module — In abstract algebra, a flat module over a ring R is an R module M such that taking the tensor product over R with M preserves exact sequences.Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are… …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Semiperfect ring — In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric. Definition Let R be ring. Then R is semiperfect if any of the following equivalent… …   Wikipedia

  • Schur's lemma — In mathematics, Schur s lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that that if M and N are two finite dimensional irreducible representations of a group G and… …   Wikipedia

  • Flora Australiens — Spinifex (Triodia) bedeckt große Gebiete des ariden Zentralaustralien …   Deutsch Wikipedia

Поделиться ссылкой на выделенное

Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»