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Local Rings, Semilocal Rings, and Idempotents

  • T. Y. Lam
Part of the Graduate Texts in Mathematics book series (GTM, volume 131)

Abstract

In the first two sections of this chapter, we focus our attention on two special classes of rings, namely, local rings and semilocal rings. By definition, a ring R is local if R/rad R is a division ring, and R is semilocal if R/rad R is a semisimple ring. Thus, local rings include all division rings, and semilocal rings include all left or right artinian rings. The basic properties of local and semilocal rings are developed, respectively, in §19 and §20. We shall see, for instance, that local rings are connected with the problem of the uniqueness of Krull-Schmidt decompositions, and that semilocal rings are connected with the problem of “cancellation” of modules.

Keywords

Local Ring Left Ideal Division Ring Endomorphism Ring Artinian 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.

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • T. Y. Lam
    • 1
  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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