Closed-coessential and Closed-coclosed Submodules

Esraa H. Yahyaa; Sahira Mahmood yaseen

doi:10.24996/ijs.2025.66.1.%g

Authors

Esraa H. Yahyaa Mathematics Department,College of Science ,University of Baghdad, Iraq https://orcid.org/0009-0009-4818-392X
Sahira Mahmood yaseen Mathematics Department,College of Science ,University of Baghdad, Iraq

DOI:

https://doi.org/10.24996/ijs.2025.66.1.%25g

Keywords:

closed-small submodule,, closed-coessential submodule,, closed-coclosed submodule

Abstract

The aim of this study is to present the concept of closed-coessential submodule and closed-coclosed submodule, the consideration of certain properties. Allow R be a ring with identity and define F as a left R-module on the left, with H and E acting as its submodules, in that sense , E≤H≤F then E is called closed- coessential submodule in E of H (E ≤_(c.ce) H ) , if H/E ≪_c F/E .

On the other hand, a submodule H of F is known as closed-coclosed submodule, if E is closed-coessential submodule of H in F. Finally, in this article we introduce some properties of these types of submodules under some conditions which are in analogy with the known properties for coessential and coclosed submodules properties. And we discuss the relation between them with the examples and remarks are needed in our work.