Sigma Cyclic and Serial Rings

Abstract

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.KeywordsLeft IdealPrime RingProjective ModuleUniserial ModuleSerial RingThese 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.