Exploring virtual versions of H-supplemented and NS-modules.

In this work we introduce virtual versions of H-supplemented modules and NS-modules. These modules are defined by replacing the condition of being a "direct summand" with being "isomorphic to a direct summand". The paper explores various equivalent conditions for a module to be virtually H-supplemented and investigates their fundamental properties. It is discovered that over a right V-ring for a module, the concepts virtually H-supplemented, virtually semisimple and VNS, coincide. Additionally, it is proven that each right R-module is VNS if and only if every noncosingular right R-module is injective. [ABSTRACT FROM AUTHOR]