Strongly-Pseudo-Extending Modules and SPmodules

Abstract

Let R be a ring and M be an R-module. Recall that M is extending if, every submodule of M is essential in a direct summand of M. And recall that an R-module M is fully pseudo stable if every submodule of M is pseudo stable. In this work, we introduce and study two classes of modules. The first class is stronger than extending modules, and the second class is generalization of fully pseudo stable modules. We call an R-module M is strongly-pseudo-extending if, every submodule of M is essential in a pseudo stable direct summand of M. We call an R-module of M is SPmodule if, every direct summand of M is pseudo stable. Many characterizations and properties of these concepts are given. Moreover, the relation among these concepts is studied. It is shown that an Rmodule M is strongly-pseudo-extending if and only if M is an extending and M is SP-module