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Projective Covers and Perfect Rings

  • Carl Faith
Chapter
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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 191)

Abstract

A morphism f: AB of R-modules is said to be minimal provided that ker f is a superfluous submodule of A. For example, for a right ideal I, the canonical map R → R/I is superfluous if and only if I ⊆ rad R 18.3. A module A is a projective cover (proj. cov.) of B provided that A is projective and there exists a minimal epimorphism A → B. This notion is dual to that of injective hull, and yet, although each R-module has an injective hull, projective covers of modules may fail to exist. For example, as is shown in this chapter, a necessary condition that every right R-module have a projective cover is that R/rad R be semisimple, and rad R be a nil ideal.

Keywords

Local Ring Left Ideal Projective Module Ring Theory Projective Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • Carl Faith
    • 1
    • 2
  1. 1.Rutgers, The State UniversityNew BrunswickUSA
  2. 2.The Institute for Advanced StudyPrincetonUSA

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