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Paper IPM / M / 8290  


Abstract:  
Let R be a commutative Noetherian Nagata ring, let M be a
nonzero finitely generated Rmodule, 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→ N_{a}
on the set of submodules N of M is a semiprime 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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