On the cofiniteness of local cohomology modules

In this note we show that if $I$ is an ideal of a Noetherian ring $R$ and $M$ is a finitely generated $R$-module, then for any minimax submodule $N$ of $H^{t}_{I}(M)$ the $R$-module ${\rm Hom}_{R}(R/I,H_{I}^{t}(M)/N)$ is finitely generated, whenever the modules $H_{I}^{0}(M),\, H_{I}^{1}(M),..., \,H_{I}^{t-1}(M)$ are minimax. As a consequence, it follows that the associated primes of $H^{t}_{I}(M)/N$ are finite. This generalizes the main result of Brodmann and Lashgari (2000).