Uniformly-S-pseudo-injective modules

This paper introduces the notion of uniformly-S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be u-S-pseudo-injective if for any submodule K of E, there is s in S such that for any u-S-monomorphism f : K → E, sf can be extended to an endomorphism g : E → E. Several properties of this notion are studied. For example, we show that an R-module M is u-S-quasi-injective if and only if M ⊕ M is u-S-pseudo-injective. New classes of rings related to the class of QI-rings are introduced and characterized.