In this paper, we introduce the notion of P-coretractable module . Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced . Also , we give the notion of strongly P-coretractable and study it comparison with P-coretractable , moreover the mono-P-coretractable concept are introduced and studied .

In this paper we introduce the concept P-dou module as a generalization of dou modules, where an R module M is called a P-duo module if every pure submodule of M is a fully invariant. Many results about this concept are given. Also some relationships between a P-dou module and other modules related with it are presented.

In this paper, we introduce two new notions related with coretrectable modules which are called essentially coretrectable and weakly essentially coretrectable modules. Some basic properties of these classes are investigated and some relationships between these modules and other related concepts are given.

Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.

In this paper we introduce and study modules with ⊛(⊛’or ⊛’’) condition. We give several properties of these types of modules and some relationships between them.