Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$,

$M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove a characterization  for ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(M)$ and ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(X,M)$ to be $\mathfrak{a}$-weakly cofinite for all $i$, whenever one of the following cases holds:

(a) ${ara} (\mathfrak{a})\leq 1$, (b) $\dim R/\mathfrak{a} \leq 1$ or (c) $\dim R\leq 2$. We also

prove that, if $M$ is a weakly Laskerian $R$-module, then ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(X,M)$ is $\mathfrak{a}$-weakly cofinite for all $i$, whenever $\dim X\leq 2$ or $\dim M\leq 2$ (resp.$(R,\mathfrak{m})$ a local ring and $\dim X\leq 3$ or $\dim M\leq 3$).  Let $d=\dim M<\infty$, we prove an equivalent condition for top local cohomology module ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{d}(M)$ to be weakly Artinian.