𝒆 ∗ -Extending Modules

This paper aims to introduce the concepts of 𝑒 ∗ -closed, 𝑒 ∗ -coclosed


Introduction
In this work  is a right module over a ring  with identity.() is the injective envelope of  .When  +  =  implies  =  for each  ≤  ,  is called a small submodule of , symbolized by  ≪ .See [1] and [2].If  ∩  ≠ {0} for each 0 ≠  ≤ , ISSN: 0067-2904 then  is called an essential submodule of , see [1] and [3].A submodule  of module  is closed if  has no proper essential extension, see [3].If every submodule of a module  is essential in the direct summand, then module is said to be extending. is an extending module if and only if each of its closed submodules is a direct summand, see [4].
As in [8], we will use  * -essential and  * -essential small submodules to present a new generalization of closed, coclosed submodules and extending modules.Namely  * -closed submodules,  * -coclosed submodules, and  * -extending modules, respectively.Moreover,we will prove the main properties of these concepts.Now, let us present the following proposition that is crucial to our work.

e*-Closed submodules
In this section, we will prove some properties of e*-closed, as introduce in [5].

Definition 2.1 [6]
A submodule  of a module  is e*-closed in , if has no proper e*-essential extension, (symbolized by  ≤  *  ).

Proposition 3.3 Let 𝑀 be a module and let
by (the second isomorphism theorem), .Therefore, 2) Suppose that  ≤  2 such that for each  ∈ .Easley sees that  is an epimorphism, so by proposition 3 in [7], , so and  2 =  +  1 .

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=  . is said to be an  *coclosed submoduleof  (symbolizedby  ≤  *  ).Examples and Remarks 3.2 1.Every  * -coclosed submodule is coclosed.Let  be a module,  be an  * -coclosed submodule of , and  a submodule of  such that Every small is  * -essential small.As a result,  is an  * -coclosed.Thus,  =  and  is a coclosed submodule of .

Proposition 4 . 6 Corollary 4 . 7 Proposition 4 . 8 A
An -module  is an e*-extending if and only if every e*-closed submodule is a direct summand.Proof.(⟹)Let be an e*-closed submodule of .Since  is e*-extending; there is a direct summand  of  with  ≤  * .But  is e*-closed.Hence,  = .(⟸) Let  be a submodule of .Then, by Proposition 2.8.an e*-closed submodule  exists with  ≤  * .By the hypothesis,  is a direct summand.Therefore,  is an e*-extending.Under isomorphism, the e*-extending module is closed.Proof.Clear using the corollary 2.6.The direct summand of the e*-extending module is e*-extending, as shown by the following proposition.direct summand of e*-extending module is e*-extending.

3 .
For every submodule  of , there is a decomposition  is a direct summand of  and  ≤  * .