Nonsingular semiperfect CS-rings
type: Item
instance of: scholarly article
title: Nonsingular semiperfect CS-rings
zbMATH Open document ID: 0920.16003
author: S. Barthwal
author: Pramod Kanwar
author: Sergio R. López-Permouth
author: S. K. Jain
published in: Journal of Algebra
publication date: 1999-07-19T00:00:00Z
review text: A ring \(R\) is called a right CS-ring if every complement right ideal of \(R\) is a direct summand. The main aim of this paper is to characterize right nonsingular semiperfect right CS-rings. The authors show that, under some additional condition, every right nonsingular semiperfect right CS-ring can be represented as a direct sum of block triangular matrix rings. This structure theorem generalizes a result on right nonsingular semiprimary right CS-rings obtained earlier by \textit{A. W. Chatters} and \textit{C. R. Hajarnavis} [Q. J. Math., Oxf. II. Ser. 28, 61-80 (1977; Zbl 0342.16023)]. The proof of this theorem is based on several interesting lemmas that are of independent interest. For example, it is shown that for a positive integer \(n>1\), the \(n\times n\) matrix ring over a local right Ore domain \(K\) is a right CS-ring if and only if \(K\) is a right and left valuation domain. The reviewer has received a preprint by \textit{S. K. Jain, P. Kanwar} and \textit{S. R. López-Permouth} [Nonsingular semiperfect CS-rings. II. Bull. Lond. Math. Soc. (to appear)], in which the authors have obtained a general structure theorem for right nonsingular semiperfect right CS-rings, without any additional conditions.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1200004
zbMATH Keywords: nonsingular rings
zbMATH Keywords: semiperfect rings
zbMATH Keywords: CS-rings
zbMATH Keywords: direct summands
zbMATH Keywords: direct sums of matrix rings
MaRDI profile type: Publication
full work available at URL: jabr.1997.7336
OpenAlex ID: W2053530230
cites work: RINGS IN WHICH EVERY COMPLEMENT RIGHT IDEAL IS A DIRECT SUMMAND
cites work: Q4843160
cites work: Q3041361
cites work: Cyclic modules whose quotients have all complement submodules direct summands
cites work: Maximal quotient rings of generalized matrix rings
DOI: 10.1006/JABR.1997.7336
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description: scientific article; zbMATH DE number 1200004