On ps-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⊆M such that M=U+V and the intersection U∩V is projective semisimple. We demonstrate that, over an arbitrary ring, the collection PS of short exact sequences of the form 0⟶M⟶ψN⟶ϕK⟶0, where Im(ψ) 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.