On <i>ps</i>-Supplement Submodules

In this work, we introduce ps-supplement submodules as a kind of supplements. Specifically, a submodule V of a module M is referred to as a ps-supplement if there exists another submodule U subset of M such that M=U+V and the intersection U boolean AND V is projective semisimple. We demonstrate that, over an arbitrary ring, the collection PS of short exact sequences of the form 0 -> M ->psi N ->phi K -> 0, where Im(psi) is a ps-supplement in N, constitutes a proper class. In addition, we explore several homological aspects of this class, including the symmetry of the relations between the modules. Finally, we provide that a ring R is a left SI-ring with essential socle if and only if every left R-module is ps-supplemented.