Finiteness properties of local cohomology modules for $\mathfrak{a}$-minimax modules

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ an ideal of $R$. In this paper we introduce the concept of $\mathfrak{a}$-minimax $R$-modules, and it is shown that if $M$ is an $\mathfrak{a}$-minimax $R$-module and $t$ a non-negative integer such that ${\rm H}_\mathfrak{a}^i(M)$ is $\mathfrak{a}$-minimax for all $i<t$, then for any $\mathfrak{a}$-minimax submodule $N$ of ${\rm H}_\mathfrak{a}^t(M)$, the $R$-module ${\rm Hom}_R(R/\mathfrak{a},{\rm H}_\mathfrak{a}^t(M)/N)$ is $\mathfrak{a}$-minimax. As a consequence, it follows that the Goldie dimension of ${\rm H}_\mathfrak{a}^t(M)/N$ is finite, and so the associated primes of ${\rm H}_\mathfrak{a}^t(M)/N$ are finite. This generalizes the main result of Brodmann and Lashgari (2000).