On Strongly Circle plus-Supplemented Modules

Abstract

Strongly circle plus-supplemented and strongly cofinitely circle plus-supplemented modules are defined and some properties of strongly circle plus-supplemented and strongly cofinitely circle plus-supplemented modules are investigated. Let R be a ring. Then every R-module is strongly circle plus-supplemented if and only if R is perfect. The finite direct sum of circle plus-supplemented modules is circle plus-supplemented. However, this is not true for strongly circle plus-supplemented modules. Any direct sum of cofinitely circle plus-supplemented modules is cofinitely circle plus-supplemented but this is not true for strongly cofinitely circle plus-supplemented modules. We also prove that a supplemented module is strongly circle plus-supplemented if and only if every supplement submodule lies above a direct summand.