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Let $R$ be a commutative Noetherian Nagata ring, let $M$ be a
non-zero finitely generated $R$-module, and let $I$ be an ideal of
$R$ such that height $_{M}I > 0$. In this paper, there is a
definition of the integral closure $N_{a}$ for any submodule $N$
of $M$ extending Rees' definition for the case of a domain. As the
main results, it is shown that the operation $N\rightarrow N_{a}$
on the set of submodules $N$ of $M$ is a semi-prime operation, and
for any submodule $N$ of $M$, the sequences Ass$_{R}M/(I^{n}
N)_{a}$ and Ass$_{R}(I^{n} M)_{a}/(I^{n} N)_{a}(n = 1,2, ... )$ of
associated prime ideals are increasing and ultimately constant for
large $n$.
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