Some results on rings whose cyclic cofaithful modules are generators (Q1376661)

A ring \(R\) is called right finitely pseudo-Frobenius (briefly, right FPF) if every finitely generated faithful right \(R\)-module is a generator. A ring \(R\) is said to be generated by faithful right cyclics (briefly, right GFC) if every cyclic faithful right \(R\)-module is a generator. GFC rings were first considered by Birkenmeier. A generalization of right self-injective and right FPF rings has been introduced and investigated by the author [Algebra-Ber. 70 (1993; Zbl 0830.16007)]: A ring \(R\) is called right FSG if every finitely generated cofaithful right \(R\)-module (=right \(R\)-subgenerator) is a generator. Now we define a class of rings which is a generalization of GFC and FSG rings: A ring \(R\) is called right CSG if every cyclic cofaithful right \(R\)-module (=right \(R\)-subgenerator) is a generator. The purpose of this work is to present a study of the class of these rings.