Sigma Cyclic and Serial Rings

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


The first three sections of this chapter present the structure of serial rings of Warfield [75]. The main theorem 25.3.4 characterizes when every finitely presented left module over a ring R is a direct sum of uniserial modules: this happens iff R is itself such a direct sum both as right and left module, that is, iff R is serial. (See Section 0 for definitions.) In this case, then for any finitely generated submodule M of a finitely generated projective module P, there are “stacked” decompositions of P and M into direct sums of uniserial modules (see 25.3.3ff). Moreover, any Noetherian serial ring is decomposable into a finite product of Artinian and (semi)prime rings (25.3.5). This is reminiscent of the theorems of Chatters (20.30) for hereditary rings, and Krull-Asano-Goldie 20.37, for principal ideal rings, and, in fact, Robson’s general method (20.35) used to prove these also applies here.


Left Ideal Prime Ring Projective Module Uniserial Module Serial Ring 
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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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