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Direct Sum Representations of Rings and Modules

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

Abstract

If C is a class of right R-modules, a module M will be called sigma C, in case M is isomorphic to a direct sum of modules in C. We will write Σ-C for short. Also, if M is a class of right R-modules which is Σ-C then we say that R is right Σ-C. For example, if is the class of (injective) right R-modules, and C is the class of finitely generated right R-modules, then the corresponding statement is that R is right (injective) Σ-finitely generated. Also, the statement R is right (injective) Σ-cyclic means that every (injective) right R-module is a direct sum of cyclic modules. (Note that one applies these designations to R as an object of RINGS rather than as an object of mod-R; strictly speaking, it is mod-R (not R) that is (injective) Σ-finitely generated when we say that R is.)

Keywords

Prime Ideal Local Ring Direct Summand Prime Ring Valuation Ring 
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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