Applying the duo and essential properties on extending modules

Abstract

A module M is called C_1-module (extending) if for all N≤M, there exists a direct summand B≤M∋ B is an essential of N in M. The main goal is to get a module namely C_1-module (extending). Meaning we look for conditions and algebraic structures that lead to obtaining the submodules to be essential, and thus we obtain the C_1-module (extending). The tools that enable us to get this goal are semi simple module, multiplication module and injective module. The second goal is to obtain a generalization of C_1-module using the following tools: duo submodule, prime and semi prime submodules (P-(S.P) submodules) and quasi-injective submodule (Q-injective). Finally; we can say that all the results in this work depended on the concept of submodules of the module M and the ring R in this thesis stands for a commutative ring with identity.