Associated primes, integral closures and ideal topologies
Volume 105 / 2006
Abstract
Let $\mathfrak{a}\subseteq \mathfrak{b}$ be ideals of a Noetherian ring $R$, and let $N$ be a non-zero finitely generated $R$-module. The set $\overline{Q}^{*}(\mathfrak{a}, N)$ of quintasymptotic primes of $\mathfrak{a}$ with respect to $N$ was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A^*_a(\mathfrak{b}, N):= \bigcup _{n\geq1}\mathop{\rm Ass}_RR/(\mathfrak{b}^n)^{(N)}_a$ of associated primes is finite. The purpose of this paper is to show that the topology on $N$ defined by $\{(\mathfrak{a}^n)_a^{(N)}:_R \langle \mathfrak{b}\rangle \}_{n\geq1 }$ is finer than the topology defined by $\{(\mathfrak{b}^n)_a^{(N)}\}_{n\geq 1}$ if and only if $A^*_a(\mathfrak{b}, N)$ is disjoint from the quintasymptotic primes of $\mathfrak{a}$ with respect to $N$. Moreover, we show that if $\mathfrak{a}$ is generated by an asymptotic sequence on $N$, then $A^*_a(\mathfrak{a}, N)= \overline{Q}^{*}(\mathfrak{a}, N)$.
