Asymptotic behaviour of integral closures of ideals relative to Artinian modules (Q1204378)

Let \(I\) be an ideal of a commutative ring \(R\) and let \(A\) be an Artinian \(R\)-module. Denote by \(I^*\) the integral closure of \(I\) relative to \(A\) as introduced by \textit{R. Y. Sharp} and \textit{A.-J. Taherizadeh} [J. Lond. Math. Soc., II. Ser. 37, No. 2, 203-218 (1988; Zbl 0656.13001)]. The structure of \(A\) over \(R\) is closely related to its structure over a certain complete semilocal Noetherian ring \(S\) [compare \textit{R.Y. Sharp}, Math. Proc. Camb. Philos. Soc. 111, No. 1, 25-33 (1992; Zbl 0758.13007)]. By setting up a Matlis duality over \(S\) and using a result of Ratliff on asymptotic primes, it is shown that, for an ideal \(J\) of \(S\), the sequence \(\text{Ass}((J^ n)^*)\) is increasing and ultimately constant. In \(R\), it follows that \(\bigcup_ n\text{Ass}((I^ n)^*)\) is finite.