Annihilators and the CS-condition

Abstract It is proved that if every cyclic right R-module is torsionless and R is a left CS-ring then R is semiperfect left continuous with soc(RR)essential in RR. As a consequence every right cogenerator, left CS-ring R is shown to be right pseudo-Frobenius and left continuous, and an example is given to show that R need not be left selfinjective. It is also proved that if R is a left CS-ring and every cyclic right R-module embeds in a free module, then R is quasi-Frobenius if and only if J(R) ⊆ Z(RR).

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