A generalization of the finiteness problem of the local cohomology modules

Let R be a commutative Noetherian ring and (mathfrak{a}) an ideal of R. We introduce the concept of (mathfrak{a}) -weakly Laskerian R-modules, and we show that if M is an (mathfrak{a}) -weakly Laskerian R-module and s is a non-negative integer such that Ext _R ^j ((R/mathfrak{a},H_mathfrak{a}^i (M))) is (mathfrak{a}) -weakly Laskerian for all i < s and all j, then for any (mathfrak{a}) -weakly Laskerian submodule X of (H_mathfrak{a}^s (M)) , the R-module (Hom_R (R/mathfrak{a},H_mathfrak{a}^s (M)/X)) is (mathfrak{a}) -weakly Laskerian. In particular, the set of associated primes of (H_mathfrak{a}^s (M)/X) is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an (mathfrak{a}) -weakly Laskerian R-module such that (H_mathfrak{a}^i (N)) (N) is (mathfrak{a}) -weakly Laskerian for all i < s, then the set of associated primes of (H_mathfrak{a}^s (M,N)) (M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).