Serial rings with right Krull dimension one. II
type: Item
instance of: scholarly article
title: Serial rings with right Krull dimension one. II
zbMATH Open document ID: 0654.16015
DOI: 10.1016/0021-8693(88)90243-8
author: Mary H. Wright
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: [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.
reviewed by: A. W. Chatters
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D30
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 4067136
zbMATH Keywords: direct sum of modules
zbMATH Keywords: serial ring of Krull dimension 1
zbMATH Keywords: Noetherian serial rings
zbMATH Keywords: direct sums of Artinian rings
zbMATH Keywords: cliques
zbMATH Keywords: simple R-modules
zbMATH Keywords: non-split extension
zbMATH Keywords: finite directed graph
zbMATH Keywords: local projective modules
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90243-8
OpenAlex ID: W2094775557
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description: scientific article; zbMATH DE number 4067136