Uniformly S-projective relative to a module and its dual

Abstract

In this article, we introduce the notion of uniformly S-projective (u-S-projective) relative to a module. Let S be a multiplicative subset of a ring R and M an R-module. An R-module P is said to be u-S-projective relative to M if for any u-S-epimorphism f : M \to N, the induced map HomR(P, f ): HomR(P, M ) \to HomR(P, N ) is a u-S-epimorphism. Dually, we also introduce u-S-injective relative to a module. Some properties of these notions are discussed. Several characterizations of u-S-semisimple modules are given in terms of these notions. The notions of u-S-quasi-projective and u-S-quasi-injective modules are also introduced, and some of their properties are discussed.