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Let $(R, \frak{m})$ be a formally equidimensional local ring of
dimension $d$. Suppose that $\Phi$ is a system of nonzero ideals
of $R$ such that, for all minimal prime ideals $\frak{p}$ of $R,
\frak{a}+\frak{p}$ is $\frak{m}$-primary for every $\frak{a}\in
\Phi$. In this paper, the main result asserts that for any ideal
$\frak{b}$ of $R$, the integral closure $\frak{b}^{*(H^d
_\Phi(R))}$ of $\frak{b}$ with respect to the Artinian $R$-module
$H^d _\Phi(R)$ is equal to $\frak{b}_\frak{a}$, the classical
Northcott-Rees integral closure of $\frak{b}$. This generalizes
the main result of [13] concerning the question raised by D. Rees.
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