Abstract

In this paper it is shown that the following two properties are equivalent over a right SC ring \(R\): (a) the direct sum of countable copies of \(R\) is a right CS module and (b) every right (and every left) module over \(R\) is CS. A corollary of this result improves on a theorem of \textit{M.-S. Chen} [in Southeast Asian Bull. Math. 24, No. 1, 25-29 (2000; Zbl 0980.16016)], which states that, given a right SC right quasi-continuous ring \(R\), “if every CS right \(R\)-module is \(\Sigma\)-CS then \(R\) is a QF ring”. The improved version (Corollary 3.2 of the paper under review) weakens the hypothesis to say only that “\(E\) is countably \(\Sigma\)-CS”. Also, a counterexample is given for Theorem 4 in [\textit{M.-S. Chen}, loc. cit.] that claims that, for a right SI right CS ring, the condition that every CS right \(R\)-module is \(\Sigma\)-CS is a direct sum of right self-injective rings. Finally, considering a property introduced in [\textit{M.-S. Chen}, loc. cit.] that for any pair of orthogonal primitive idempotents \(f\) and \(g\) in \(R\) \(fRg=0\) if and only if \(gRf=0\), it is shown that this condition reduces to being semisimple Artinian if one assumes that the ring is left perfect and right nonsingular.