Characterizing $S$-projective modules and $S$-semisimple rings by uniformity

Abstract

Let R be a ring and S a multiplicative subset of R. An R-module P is called S-projective provided that the induced sequence 0 → Hom R (P, A) → Hom R (P, B) → Hom R (P, C) → 0 is S-exact for any S-short exact sequence 0 → A → B → C → 0. Some characterizations and properties of S-projective modules are obtained. The notion of S-semisimple modules is also introduced. A ring R is called an S-semisimple ring provided that every free R-module is S-semisimple. Several characterizations of S-semisimple rings are provided by using S-semisimple modules, S-projective modules, S-injective modules and S-split S-exact sequences.