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Let $A$ be a non-zero Artinian $R$-module. For an arbitrary ideal
$I$ of $R$, we show that the local homology module $H^x_p(A)$ is
independent of the choice of {\bf{x}} whenever $0:_A I=0
:_A(x_1,\cdots , x_r)$. Thus, identifying these modules, we write
$H_p^I (A)$. In this paper we prove that there is a certain
duality between $H_i^I(A)$ and the local cohomology modules and
provide some information about the vanishing local homology module
$H_i^I(A)$ which gives an improved form of the main results of
[22].
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