Serial rings with right Krull dimension one. II
[Part I cf. ibid. 109, No.2, 319-333 (1987; Zbl 0621.16025).] A ring R is serial if, on both the left and the right, it is a direct sum of modules in each of which any two submodules are comparable. Let R be a serial ring of Krull dimension 1 (this class properly includes the Noetherian serial rings which were shown by Warfield to be direct sums of Artinian rings and prime rings). This paper studies the structure of R according to the nature of its cliques as defined below. The results themselves are too complicated to state here. Let S and T be simple R- modules. Then S is called a successor of T if there is a non-split extension of S by T. By using this definition it is possible to regard the set of isomorphism types of simple R-modules as being a finite directed graph. Each point in the graph has at most one arrow coming into it and at most one arrow going out of it. The connected components of the graph are either circular or linear in the obvious sense. The author defines a set of non-isomorphic local projective modules to be a clique if their simple factors form a connected component of the graph, and a clique is said to be circular or linear according to the nature of the corresponding connected component. The study of R is reduced to the ``elementary'' case (i.e. only one clique), which is then subdivided into various cases according to the nature of the clique.
scientific article; zbMATH DE number 4067136
scientific article