Cofiniteness of weakly Laskerian local cohomology modules

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\le n$ and we show it by $\mathrm{F}{\mathrm{D}}_{\le \mathrm{n}}$ where $n\ge -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(orminimax)submoduleNof$H^t_I(M)$theR-modules${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H_I^0(M)$, $H_I^1(M)$, …, $H_I^{t-1}(M)$ are $\mathrm{F}{\mathrm{D}}_{\le 1}$ (or weakly Laskerian). As a consequence, it follows that the associated primes of $H_I^t(M)\mathrm{/}N$ are finite. This generalizes the main results of Bahmanpour and Naghipour, Brodmann and Lashgari, Khashyarmanesh and Salarian, and Hong Quy. We also show that the category ${\mathcal{F}\mathcal{D}}^1(R,I{)}_{cof}$ of $I$-cofinite $\mathrm{F}{\mathrm{D}}_{\le 1}$ ~ $R$-modules forms an Abelian subcategory of the category of all $R$-modules.