The main goal of this paper is to introduce and study a new concept named d^*-supplemented which can be considered as a generalization of W- supplemented modules and d-hollow module. Also, we introduce a d^*-supplement submodule. Many relationships of d^*-supplemented modules are studied. Especially, we give characterizations of d^*-supplemented modules and relationship between this kind of modules and other kind modules for example every d-hollow (d-local) module is d^*-supplemented and by an example we show that the converse is not true.

The main goal of this paper is to introduce and studya new concept named Ᵹ*-supplemented which can be considered as a generalization of W-supplemented modules and Ᵹ-hollow module. Also, we introduce a Ᵹ*-supplement submodule.Many relationshipsof Ᵹ*-supplemented modules are studied. Especially, we give characterizations of Ᵹ*-supplemented modules and relationship between this kind of modules and other kind modules for example every Ᵹ-hollow (Ᵹ-local) module is Ᵹ*-supplemented and by an example we show that the converse is not true.

      In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules.  We also give some results and properties of this new kind of modules.

Let P be a right R-module, where R is a ring with identity. In the present research, a class of modules is described similar to H-μ-supplemented and μ-lifting modules. A module P is called principally H-μ-supplemented if there is a summand L of P such that pR is μ-equivalent to L for every p in P. Additionally, we present an extension of supplemented modules. A module P is considered to be principally μ-supplemented if, for every p in P, pR has a μ-supplement in P. A number of characteristics of these modules are shown, and it is demonstrated that both the Pμ-H-supplemented and Pμ-supplemented modules include the class of principally μ-lifting modules.

A non-zero submodule N of M is called essential if N L for each non-zero submodule L of M. And a non-zero submodule K of M is called semi-essential if K P for each non-zero prime submodule P of M. In this paper we investigate a class of submodules that lies between essential submodules and semi-essential submodules, we call these class of submodules weak essential submodules.