Noetherianness and Local Cohomology Modules

Let R be a commutative Noetherian ring, a an ideal of R, and M an R-module. We show that, whenever M = U-t>0(0 :(M) a(t)), M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/aN and boolean AND(t>0) a(t) N are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules H-a(n) (M) in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.