The purpose of this paper is to introduce new concepts in a module  over a ring , the first one is called e*- small essential submodule, which is a generalization of the small submodule, the second concept is called e*-radical submodule which is a generalization of the radical submodule and the last concept is called e* - hollow module which is a generalization of the hollow module. We will prove some properties of all these concepts.
The purpose of this paper is to introduce a new concept in a module M over a ring R, this concept is called e∗-essential submodule, which is a generalization of an essential submodule. We will introduce some examples and properties about this concept such that, what is the inverse image of e∗-essential submodule, the intersection of e∗-essential submodules and direct sum of e∗-essential submodules. We will show the relationship between e∗-essential submodule and Noetherian R-module. Also we will define e∗-closed submodule with some properties
The novel ideas in module M over a ring R are introduced in this study. The first one, a generalization of the e∗-lifting module, is known as e∗-hollow-lifting. The second idea, an inference of e∗-lifting, is known as a cofinite e∗-lifting module. We shall demonstrate some of these ideas’ properties
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