On generalized Q.F.D. modules

Abstract

A right R-module M is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [25], Theorem 3. In particular, it is shown that a module M is g.q.f.d. iff every direct sum of M-singular M-injective modules in σ[M] is weakly injective iff every direct sum of M-singular weakly tight is weakly tight iff every direct sum of the injective hulls of M-singular simples is weakly R-tight.