Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let a be an ideal of R and M denote the intersection of all prime ideals p is an element of Supp(R) H-a(d)(M). It is shown that H-d(a)(M) similar or equal to H-M(d)(M)/Sigma(n is an element of N) < M > (0: (HMd(M))a(n)), where for an Artinian R-module A we put (M) A = boolean AND M-n is an element of N(n) A. As a consequence, it is proved that for all ideals a of R, there are only finitely many non-isomorphic top local cohomology modules H-a(d)(M) having the same support. An addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings that need not be local.