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.
Closed-Supplemented Module
small submodules
closed submodule
closed-small submodule
closed-supplemented module
closed-weakly supplemented module