On the structure of nonsingular quasi-continuous and CS-rings

This dissertation deals with the study of three kinds of rings: (i) rings over which proper cyclic right modules are continuous, (ii) right hereditary right quasi-continuous rings, and (iii) right nonsingular semiperfect right CS-rings. It is shown that simple rings over which proper cyclic right modules are continuous (called right PCC-rings) are precisely the simple right PCI-rings and hence the simple right PCQI-rings. A Characterization of right hereditary right quasi-continuous rings is given as a direct sum of simple artinian rings and right hereditary right noetherian domains. Right nonsingular semiprimary right CS-rings have been characterized earlier by Chatters and Hajarnavis. Barthwal et. al. characterized right nonsingular semiperfect right CS-rings satisfying an additional condition (*). In this dissertation, the additional condition is removed and a complete characterization of right non-singular semiperfect right CS-rings is provided. It is shown that such rings are direct sums of certain blocked matrix rings. The Characterizations of right nonsingular semiprimary right CS-rings as obtained by Chatters and Hajarnavis and of right nonsingular semiperfect right CS-rings under the condition (*) in Barthwal et. al follow as particular cases of this structure theorem. It also follows that right nonsingular semiperfect right CS-rings with nil Jacobson radical are semiprimary.