Let a be an ideal of a commutative Noetherian ring R, M a finitely
generated R-module with finite projective dimension and N an arbitrary R-module with finite Cohen-Macaulay injective dimension. In
this paper, we show that the generalized local cohomology Hi
a(M, N)
is zero for every i larger than the Cohen-Macaulay injective dimension
of N. As applications, we obtain new characterizations of Gorenstein
and regular local rings.