Rings whose class of projective modules is socle fine

Abstract

A class C of modules over a unitary ring is said to be socle fine if whenever M, N ∈ C with Soc(M ) ∼ = Soc(N ) then M ∼ = N . In this work we characterize certain types of rings by requiring a suitable class of its modules to be socle fine. Then we study socle fine classes of quasi-injective, quasi-projective and quasicontinuous modules which we apply to find socle fine classes in special types of noetherian rings. We also initiate the study of those rings whose class of projective modules is socle fine.