Closed-Supplemented Module

Abstract

The purpose of this study is to introduce a new idea for module Q over ring R, let Q be well a unitary left R-module and let R be any ring with one. We refer to Q as a closed-supplemented module, if each submodule of Q has closed-supplement submodule where  X is called the closed-supplement submodule of C in Q ,if  Q=X+C and X∩C≪_c  X , C⊆Q .On the other hand we introduce the concept of closed-weakly supplemented module, whenever that each submodule within Q has closed-weak supplement in Q , such that a submodule X of Q is named closed-weak supplement of C , if Q=X+C and X∩C≪_c Q.We prove some properties of these class of modules. Many Several of about these ideas are given, additionally the relations involving these notions and other modules related with them are presented.