Artinian and non-artinian local cohomology modules

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\text{fa}$ and $\text{fb}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\text{fa},\text{fb}$, $\text{fa}\cap \text{fb}$ and $\text{fa}+\text{fb}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\text{fa}$ such that all local cohomology modules of $M$ with respect to $\text{fa}$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$,anymaximalelement$\fq$ofthenon-emptysetofideals${\fa$:$\H_\fa^i(M)$isnotartinianforsome$i$,$i\geq r$$\}$ is a prime ideal and that all Bass numbers of \H_\fq^i(M) are finite for all $i\ge r$.