Rad-⊕-Supplemented Modules and Cofinitely Rad-⊕-Supplemented Modules

A module M is called (cofinitely) Rad-circle plus-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-circle plus-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-circle plus-supplemented modules and Rad-circle plus-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-circle plus-supplemented module is (cofinitely) Rad-circle plus-supplemented.