  Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals
$\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$
with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$
is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an
ideal $\fa$ such that all local cohomology modules of $M$ with respect to $\fa$
have finite lengths. Also, when $r$ is an integer such that $0\leq r<
\dim_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ :
$\H_\fa^i(M)$ is not artinian for some $i$, $i\geq r$$\}$ is a prime ideal and
that all Bass numbers of $\H_\fq^i(M)$ are finite for all $i\geq r$.
