A submodule Ϝ of an R-module Ε is called small in Ε if whenever  , for some submodule W of Ε , implies  . In this paper , we introduce the notion of Ζ-small submodule , where a proper submodule Ϝ of an R-module Ε is said to be Ζ-small in Ε if  , such that  , then  , where  is the second singular submodule of Ε . We give some properties of Ζ-small submodules . Moreover , by using this concept , we generalize the notions of hollow modules , supplement submodules, and supplemented modules into Ζ-hollow modules, Ζ-supplement submodules, and Ζ-supplemented modules. We study these concepts and provide some of their relations .

The main goal of this paper is introducing and studying a new concept, which is named H-essential submodules, and we use it to construct another concept called Homessential modules. Several fundamental properties of these concepts are investigated, and other characterizations for each one of them is given. Moreover, many relationships of Homessential modules with other related concepts are studied such as Quasi-Dedekind, Uniform, Prime and Extending modules.

    Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman studied Dual Rickart modules. The main purpose of this paper is to define strong dual Rickart module. Let M and N be R- modules , M is called N- strong dual Rickart module (or relatively sd-Rickart to N)which is  denoted by M it is N-sd- Rickart if for every submodule A of M and every homomorphism fHom (M , N) , f (A) is a direct summand of N. We prove that for an R- module M , if R is M-sd- Rickart , then every cyclic submodule of M is a direct summand . In particular, if M<

We introduce in this paper the concept of an approximately pure submodule as a     generalization of a pure submodule, that is defined by Anderson and Fuller. If every submodule of an R-module  is approximately pure, then  is called F-approximately regular. Further, many results about this concept are given.

The concept of St-Polyform modules, was introduced and studied by Ahmed in [1], where a module M is called St-polyform, if for every submodule N of M and for any homomorphism 𝑓:N M; ker𝑓 is St-closed submodule in N. The novelty of this paper is to dualize this class of modules, the authors call it CSt-polyform modules, and according to this dualizations, some results which appeared in [1] are dualized for example we prove that in the class of hollow modules, every CSt-polyform module is coquasi-Dedekind. In addition, several important properties of CSt-polyform module are established, and other characterization of CSt-polyform is given. Moreover, many relationships of CSt-polyform modules with other related concepts are

In this paper we study the concepts of δ-small M-projective module and δ-small M-pseudo projective Modules as a generalization of M-projective module and M-Pseudo Projective respectively and give some results.

In this paper we study the concepts of δ-small M-projective module and δ-small M-pseudo projective Modules as a generalization of M-projective module and M-Pseudo Projective respectively and give some results.

We introduce the notion of t-polyform modules. The class of t- polyform modules contains the class of polyform modules and contains the class of t-essential quasi-Dedekind.

     Many characterizations of t-polyform modules are given. Also many connections between these class of modules and other types of modules are introduced.

In this paper, we introduce the concept of e-small Projective modules as a generlization of Projective modules.

The present study introduces the concept of J-pure submodules as a generalization of pure submodules. We  study some of its basic  properties  and  by using this concept we  define the class of  J-regular modules,  where an R-module  M is called  J-regular module if every submodule of M is J-pure submodule. Many results about this concept are proved