Prime virtually semisimple modules and rings

This article is a sequel to the recent three papers on "virtually semisimple modules and rings," by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if (resp., R-R) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or -virtually) semisimple modules. A ring R is called left (resp., right) -virtually semisimple if (resp., R-R) is -virtually semisimple. The results of the article are inspired by a characterization of left -virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case , where each D-i is a domain and each is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is -virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable -virtually semisimple modules? Finally, we study -virtually semisimple modules over commutative rings.