### A generalization of the finiteness problem of the local cohomology modules

Let $R$ be a commutative Noetherian ring and $\U0001d51e$ an ideal of $R$. We introduce the concept of $\U0001d51e$-weakly Laskerian $R$-modules, and we show that if $M$ is an $\U0001d51e$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\mathrm{Ext}}_{R}^{j}(R/\U0001d51e,{H}_{\U0001d51e}^{i}\left(M\right))$ is $\U0001d51e$-weakly Laskerian for all $i<s$ and all $j$, then for any $\U0001d51e$-weakly Laskerian submodule $X$ of ${H}_{\U0001d51e}^{s}\left(M\right)$, the $R$-module ${\mathrm{Hom}}_{R}(R/\U0001d51e,{H}_{\U0001d51e}^{s}\left(M\right)/X)$ is $\U0001d51e$-weakly Laskerian. In particular, the set of associated primes of ${H}_{\U0001d51e}^{s}\left(M\right)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an $\U0001d51e$-weakly...