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Let $(R, \frak{m})$ be a commutative Noetherian local ring and let
$\frak{a}$ be an ideal of $R$. We give some inequalities between
the Bass numbers of an $R$-module and those of its local
cohomology modules with respect to $\frak{a}$. As an application
of these inequalities, we recover results of Delflno-Marlcy and
Kawasaki by showing that for a minimax $R$-module $M$ and for any
non-negative integer $i$, the Bass numbers of the $i$th local
cohomology module $H^{i}_\frak{a}(M)$ are finite if one of the
following holds:\noindent(a) dim $R/\frak{a} = 1$,\(b) $\frak{a}$ is a principal ideal.
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