In this work, we introduce a new generalization of both Rationally extending and Goldie extending which is Goldie Rationally extending module which is known as follows: if for any submodule K of an R-module M there is a direct summand U of M (denoted by  U⊆_⊕ M) such that K β_r  U. A β_r  is a relation of K⊆M and U⊆M, which defined as  K β_r  U if and only if  K ⋂U⊆_r K and K⋂U⊆_r U.

Let R be a commutative ring with 1 and M be a (left) unitary R – module. This essay gives generalizations for the notions prime module and some concepts related to it. We termed an R – module M as semi-essentially prime if annR (M) = annR (N) for every non-zero semi-essential submodules N of M. Given some of their advantages characterizations and examples, and we study the relation between these and some classes of modules.

In this paper, we introduce a type of modules, namely S-K-nonsingular modules, which is a generalization of K-nonsingular modules. A comprehensive study of these classes of modules is given.

        Some authors studied modules with annihilator of every nonzero submodule is prime, primary or maximal. In this paper, we introduce and study annsemimaximal and coannsemimaximal modules, where an R-module M is called annsemimaximal (resp. coannsemimaximal) if ann_RN (resp. ) is semimaximal ideal of R for each nonzero submodule N of M.

  Let R be an associative ring with identity and M a non – zero unitary R-module.In this paper we introduce the definition of purely co-Hopfian module, where an R-module M is said to be purely co-Hopfian if for any monomorphism f Ë› End (M), Imf is pure in M and we give  some properties of this kind of modules.

In this paper we introduce G-Rad-lifting module as aproper generalization of lifting module, some properties of this type of modules are investigated. We prove that if M is G-Rad- lifting and
, then
, and
are G-Rad- lifting, hence we Conclude the direct summand of G-Rad- lifting is also G-Rad- lifting. Also we prove that if M is a duo module with
and
are G- Rad- lifting then M is G-Rad- lifting.

Let M be an R-module. In this paper we introduce the concept of quasi-fully cancellation modules as a generalization of fully cancellation modules. We give the basic properties, several characterizations about this concept. Also, the direct sum and the localization of quasi-fully cancellation modules are studied.