On the finiteness of Bass numbers of local cohomology modules and Cominimaxness

In this paper, we continue the study of cominimaxness modules with respect to an ideal of a commutative Noetherian ring (cf. \cite{ANV}), and Bass numbers of local cohomology modules. Let $R$ denote a commutative Noetherian local ring and $I$ an ideal of $R$. We first show that the Bass numbers ${\mu }^0(\mathfrak{p},H_I^2(R))$ and $\mu^1(\frak p, H^2_I®)$arefiniteforall$\frak p\in \Spec R$,whenever$R$ is regular. As a consequence, it follows that the Goldie dimension of $H_I^2(R)$ is finite. Also, for a finitely generated $R$-module $M$ of dimension $d$, it is shown that the Bass numbers of $H_I^{d-1}(M)$ are finite if and only if ${\text{Ext}}_R^i(R\mathrm{/}I,H_I^{d-1}(M))$ be minimax for all $i\ge 0$. Finally, we prove that if $\dim R\mathrm{/}I=2$, then the Bass numbers of $H_I^n(M)$ are finite if and only if ${\text{Ext}}_R^i(R\mathrm{/}I,H_I^n(M))$ be minimax, for all $i\ge 0$, where $n$ is a non-negative integer.