Asymptotic Behavior of Integral Closures in Modules.

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 heightMI > 0. In this paper, there is a definition of the integral closure Na 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 → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences AssR M/(InN)a and AssR (InM)a/(InN)a(n=1,2,...) of associated prime ideals are increasing and ultimately constant for large n. [ABSTRACT FROM AUTHOR]