ON A GENERALIZATION OF CQF-3′ MODULES AND COHEREDITARY TORSION THEORIES

Throughout this paper we assume that R is a right perfect ring with identity and let Mod-R be the category of right R-modules. Let M be a right R-module. We denote by 0 → K(M) → P(M) → M → 0 the projective cover of M. M is called a CQF-3′ module, if P(M) is M-generated, that is, P(M) is isomorphic to a homomorphic image of a direct sum ⊕M of some copies of M. A subfunctor of the identity functor of Mod-R is called a preradical. For a preradical σ, T_σ := {M ∈ Mod-R : σ(M) = M} is called the class of σ-torsion right R-modules, and F_σ := {M ∈ Mod-R : σ(M) = 0} is called the class of σ-torsionfree right R-modules. A right R-module M is called σ-projective if the functor Hom_R(M,−) preserves the exactness for any exact sequence 0 → A → B → C → 0 with A ∈ F_σ. We put P_σ(M) = P(M)/σ(K(M)) for a module M. We call a right R-module M a σ-CQF-3′ module if P_σ(M) is M-generated. In this paper, we characterize σ-CQF-3′ modules and give some related facts.