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Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that where for an Artinian R-module A we put A. As a consequence, it is proved that for all ideals of R, there are only finitely many non-isomorphic top local cohomology modules having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings that need not be local.