Rad-⊕-Supplemented Modules

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad- ⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P*) if and only if R is an artinian serial ring and J² = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.