Ps-coinjective modules and PS-supplement submodules

In this paper, introduced ps-supplement submodules and characterizations of some ring classes were obtained. A submodule H of a module M is called ps-supplement in M if there exists a submodule G of M such that M=G+H and G∩H is projective semisimple. Over an arbitrary ring the class PS of all short exact sequences 0⟶M□(→┴ψ ) S→┴ϕ T⟶0 such that Im(ψ) is a ps-supplement in S is a proper class. This proper class was denoted by PS. Some homological objects of this proper class have been studied. A module M is called PS-coinjective if every short exact sequence of left R-modules 0⟶M□(→┴ψ ) S→┴ϕ T⟶0 starting with the module M is in the proper class PS. A ring R is a left SI-ring with essential socle if and only if every left R-module is PS-coinjective.