On QF rings and artinian principal ideal rings

In this work we give sufficient conditions for a ring R to be quasi-Frobenius, such as R being left artinian and the class of injective cogenerators of R-Mod being closed under projective covers. We prove that R is a division ring if and only if R is a domain and the class of left free R-modules is closed under injective hulls. We obtain some characterizations of artinian principal ideal rings. We characterize the rings for which left cyclic modules coincide with left cocyclic R-modules. Finally, we obtain characterizations of left artinian and left coartinian rings.