On PS-Injective Modules

In this paper, we introduce the concept of ps-injective modules, which plays a key role in our approach. We develop the various properties of these modules. In particular, we prove that for a ring R, every projective semisimple R-module is injective if and only if every R-module is ps-injective. Furthermore, we show that every ps-injective R-module is injective if and only if R is a left semiartinian and left SI-ring. In addition, using the projective socle of the ring R, we investigate in detail the structure of modules whose socle is singular.