Rings with many direct summands
type: Item
instance of: scholarly article
title: Rings with many direct summands
zbMATH Open document ID: 0726.16017
DOI: 10.1007/BF01200342
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let R be a ring, and let X be a class of right R-modules which contains the zero module, is closed under isomorphism, and is such that every module in X has finite uniform dimension. The author investigates the situation in which every cyclic right R-module is the direct sum of a projective module and a module in the class X. A characterisation of such rings with no non-zero right ideals in the class X is given in terms of triangular matrices. This is then specialised to the case in which R is right Noetherian and X contains all simple right R-modules. In particular it is shown that every cyclic right R-module is the direct sum of a projective module and a module of finite length if and only if R is right Noetherian and has a two-sided ideal I with the following properties: \(I=eR\) for some idempotent e; I is Artinian as a right R-module; and the ring R/I is a direct sum of prime right CS rings with right Krull dimension 1.
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 4198236
zbMATH Keywords: uniform dimension
zbMATH Keywords: cyclic right R-module
zbMATH Keywords: direct sum
zbMATH Keywords: right Noetherian
zbMATH Keywords: simple right R-modules
zbMATH Keywords: projective module
zbMATH Keywords: module of finite length
zbMATH Keywords: direct sum of prime right CS rings
zbMATH Keywords: right Krull dimension
author: Patrick F. Smith
reviewed by: A. W. Chatters
MaRDI profile type: Publication
cites work: RINGS IN WHICH EVERY COMPLEMENT RIGHT IDEAL IS A DIRECT SUMMAND
cites work: Semi-Prime Rings with Maximum Condition
cites work: Krull dimension
cites work: On rings with restricted minimum condition
cites work: SOME RINGS WHICH ARE CHARACTERISED BY THEIR FINITELY GENERATED MODULES
description: scientific article; zbMATH DE number 4198236