Characterizations of Rad-supplemented modules

We prove that a commutative ringR is an artinian principal ideal ring if and only if the ring is semilocal and every Rad-supplemented R-module is a direct sum of w-local R-modules. Moreover, we study of extensions of Rad-supplemented modules over commutative noetherian rings, and we show that if M N is reduced, M is Rad-supplemented if and only if N and M N are Rad-supplemented. We also prove that over a dedekind domain an indecomposable, amply Rad-supplemented radical module is hollow radical. 2000 Mathematics Subject Classification: 16G10; 16D10; 16D99