Rad-?-supplemented modules

Abstract

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.