On Goldie-supplemented modules

Let M be a left R-module. X,Y of M are ß· equivalent, Xß·Y , if and only if ... is small in ... and is small in ... . A module M is called G·-supplemented if for every submodule X of M there is a supplement submodule S of M such that Xß·S. In this work some new properties of ß· are given and G·-supplemented modules are studied. Also completely G·-supplemented modules and G· -radical supplemented modules are defined.