COFINITELY I-RAD- ⊕-SUPPLEMENTED MODULES

Let M be an R-module and let I be an ideal of R. We say that M is cofinitely I-Rad- ⊕-supplemented modules, provided for every cofinite submodule N of M, there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K ⊆Rad(K). The aim of this paper is to show new properties of cofinitely I-Rad- ⊕-supplemented modules. We show that cofinitely I-Rad- ⊕- supplemented modules is closed under finite direct sums. In addition, we prove that an R-module M is cofinitely I-Rad- ⊕-supplemented if and only if K and M/K are cofinitely I-Rad- ⊕-supplemented for an ideal I of R and cofinite fully invariant direct summand K of M.