Nonsingular semiperfect CS-rings
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.
scientific article; zbMATH DE number 1200004