On rings whose cyclic faithful modules are generators (Q1911581)

A ring \(R\) is called right FPF (resp. right GFC) if every finitely generated (resp. cyclic) faithful right \(R\)-module is a generator for the category of right \(R\)-modules. It is shown that the maximal right quotient ring of a right nonsingular right GFC ring is a left and right FPF ring. Also quasi-Baer right GFC rings, right p.p. GFC rings, and right p.p. quasi-Baer right GFC rings are characterized together with several interesting examples. Furthermore, it is proved that a (von Neumann) regular ring \(R\) is right GFC if and only if \(R\) is isomorphic to a finite direct product of an abelian regular ring and full matrix rings over self-injective abelian regular rings.