Closed-coessential and Closed-coclosed Submodules - Yahyaa, Esraa H.; yaseen, Sahira Mahmood (Iraqi Journal of Science, 2025)
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.