ON A GENERALIZATION OF QF-3′ MODULES AND HEREDITARY TORSION THEORIES

Let R be a ring with identity, and let Mod-R be the category of right R-modules. Let M be a right R-module. We denote by E(M) the injective hull of M. M is called QF-3′ module, if E(M) is M-torsionless, that is, E(M) is isomorphic to a submodule of a direct product ΠM 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 the class of σ-torsion right R-modules, and F_σ := {M ∈ Mod-R : σ(M) = 0} is the class of σ-torsionfree right R-modules. A right R-module M is called σ-injective if the functor Hom_R(−,M) preserves the exactness for any exact sequence 0 → A → B → C → 0 with C ∈ T_σ. A right R-module M is called σ-QF-3′ module if E_σ(M) is M-torsionless, where E_σ(M) is defined by E_σ(M)/M := σ(E(M)/M). In this paper, we characterize σ-QF-3′ modules and give some related facts.