Relative QF-3' modules and QF-3 modules

Abstract

As a generalization of QF rings, the notion of rings was introduced. Many authors have studied connections between rings, Morita duality and quotient rings. Basic part of these works can be found in [21]. Several results for rings are extended to more general classes of rings. Rings R whose injective hull E(RR) is torsionless (right QF-3' rings) and rings R in which every finitely generated submodule of the injective hull E(RR) is torsionless (according to (91, we call such rings right QF-3 rings) are such kind of rings. These rings are closely related to the dual functors, the Lambek torsion theories, and Morita duality for Grothendieck categories (see [5], [6], [7], [9] and [ID]). On the other hand, these rings are generalized to modules, as QF-3' and QF-3 modules. QF-3' modules were mainly investigated from torsion theoretical aspects in [2] and [13]. Recently QF3 modules were introduced and studied by Ohtake in [16]. He investigated a relationship between QF-3 modules and localized Morita dualities. In this paper we shall introduce and investigate relative QF-3' modules and modules. Relative QF-3' modules are generalizations of QF-3'