Uniformly S-pseudo-projective modules

In this paper, we introduce the notion of uniformly S-pseudo-projective (u-S-pseudo-projective) modules as a generalization of u-S-projective modules. Let R be a ring and S a multiplicative subset of R. An R-module P is said to be u-S-pseudo-projective if for any submodule K of P, there is s\in S such that for any u-S-epimorphism f:P\to \frac{P}{K}, sf can be lifted to an endomorphism g:P\to P. We prove that an R-module M is u-S-quasi-projective if and only if M\oplus M is u-S-pseudo-projective. Also, we prove that if A\oplus B is u-S-pseudo-projective, then any u-S-epimorphism f:A\to B u-S-splits. Several new characterizations of u-S-semisimple rings are given.