Throughout this work we introduce the notion of Annihilator-closed submodules, and we give some basic properties of this concept. We also introduce a generalization for the Extending modules, namely Annihilator-extending modules. Some fundamental properties are presented as well as  we discuss the relation between this concept and some other related concepts.

Let R be any ring with identity, and let M be a unitary left R-module. A submodule K of M is called generalized coessential submodule of N in M, if Rad( ). A module M is called generalized hollow-lifting module, if every submodule N of M with is a hollow module, has a generalized coessential submodule of N in M that is a direct summand of M. In this paper, we study some properties of this type of modules.

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

An -module  is extending if every submodule of   is essential in a direct summand of . Following Clark, an -module  is purely extending if every submodule of   is essential in a pure submodule of . It is clear purely extending is generalization of extending modules. Following Birkenmeier and Tercan, an -module     is Goldie extending if, for each submodule      of , there is a direct summand D of such that . In this paper, we introduce and study class of modules which are proper generalization of both the purely extending modules and -extending modules. We call an -module  is purely Goldie extending if, for each , there is a pure submodule P of such that  . Many c

A Module M is called cofinite  J- Supplemented  Module  if for every  cofinite submodule L of  M, there exists a submodule N of M such that M=L+N with   main properties of cof-J-supplemented modules.  An R-module M is called fully invariant-J-supplemented if for every fully invariant submodule N of M, there exists a submodule K of M, such that M = N + K with N  K K. A condition under which the direct sum of FI-J-supplemented modules is FI-J-supplemented was given. Also, some types of modules that are related to the FI-J-supplemented module were discussed.

 In this paper, we give a comprehensive study of min (max)-CS modules such as a closed submodule of min-CS module is min-CS. Amongst other results we show that a direct summand of min (max)-CS module is min (max)-CS module. One of interested theorems in this paper is, if R is a nonsingular ring then R is a max-CS ring if and only if R is a min-CS ring.

      Fuchs introduced purely extending modules as a generalization of extending modules. Ahmed and Abbas gave another generalization for extending modules named semi-extending modules. In this paper, two generalizations of the extending modules are combined to give another generalization. This generalization is said to be almost semi-extending. In fact, the purely extending modules lies between the extending and almost semi-extending modules. We also show that an almost semi-extending module is a proper generalization of purely extending. In addition, various examples and important properties of this class of modules are given and considered. Another characterization of almost semi-extending modules is established. Moreover, the re

Let R be a commutative ring with identity 1 ¹ 0, and let M be a unitary left module over R. A submodule N of an R-module M is called essential, if whenever N ⋂ L = (0), then L = (0) for every submodule L of M. In this case, we write N ≤e M. An R-module M is called extending, if every submodule of M is an essential in a direct summand of M. A submodule N of an R-module M is called semi-essential (denoted by N ≤sem M), if N ∩ P ≠ (0) for each nonzero prime submodule P of M. The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules, where a submodule N of

   Let R be a ring with identity and Ą a left R-module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R-module A is s-compressible if for any nonzero submodule B of A there exists a small f in Hom_R(A, B). An R-module A is s-prime if for any submodule B of A, ann_R (B) A is small in A. These concepts and related concepts are studied in as well as  many results consist properties and characterizations are obtained.  

In this paper, we introduce a new concept named St-polyform modules, and show that the class of St-polyform modules is contained properly in the well-known classes; polyform, strongly essentially quasi-Dedekind and ?-nonsingular modules. Various properties of such modules are obtained. Another characterization of St-polyform module is given. An existence of St-polyform submodules in certain class of modules is considered. The relationships of St-polyform with some related concepts are investigated. Furthermore, we introduce other new classes which are; St-semisimple and ?-non St-singular modules, and we verify that the class of St-polyform modules lies between them.