Totally Goldie*-Supplemented Modules

Abstract

In this paper, we first consider the properties of the Goldie*-supplemented modules, and we study the properties of totally Goldie*-supplemented modules as a version of the Goldie*-supplemented modules. A module M is called Goldie*-supplemented module if, for every submodule U of M, there exists a supplement submodule S of M such that Uβ∗S. A module M is called a totally Goldie*-supplemented module if, for every submodule A of M, A is a Goldie*-supplemented module. We emphasize that if M is totally Goldie*-supplemented, then MU is totally Goldie*-supplemented for some small submodule U of M. In addition, M=A⊕B is totally Goldie*-supplemented if A and B are totally Goldie*-supplemented. Furthermore, we mention the connection between totally Goldie*-supplemented, totally supplemented, and Goldie*-supplemented.