  Let R be a (not necessarily local) Noetherian ring and M a finitely generated
R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the
intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that
H^d_{\fa}(M)\simeq H^d_{\fM}(M)/\displaystyle{\sum_{n\in
\mathbb{N}}}<\fM>(0:_{H^d_{\fM}(M)}\fa^n), where for an Artinian R-module A we
put <\fM>A=\cap_{n\in \mathbb{N}} \fM^nA. As a consequence, it is proved that
for all ideals \fa of R, there are only finitely many non-isomorphic top local
cohomology modules H^d_{\fa}(M) having the same support. In addition, we
establish an analogue of the Lichtenbaum-Hartshorne Vanishing Theorem over
rings that need not be local.
