Circle Plus-Cofinitely Supplemented Modules

Abstract

Let R be a ring and M a right R-module. M is called circle plus-cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the circle plus-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of circle plus-cofinitely supplemented modules is circle plus-cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is circle plus-cofinitely supplemented. In addition, if M has the summand sum property, then M is circle plus-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.