On the prime divisors of IJ when I is integrally closed (Q1100235)

It is shown that if \(I\subseteq P\) are ideals in a Noetherian ring \(R\) such that \(P\) is prime, and if \(P\in \text{Ass}(R/I_ a)\), then \(P\in \text{Ass}(R/IJ_ a)\cap \text{Ass}(R/(IJ)_ a)\) for all ideals \(J\) in \(R\) such that \(\text{height}(J)\geq 1\). (Here, \(K_ a\) denotes the integral closure of the ideal \(K\).) Two of the important known results for asymptotic prime divisors are special cases of this, namely: (a) if \(\text{height}(I)\geq 1\) and if \(P\in \text{Ass}(R/(I^ m)_ a)\) for some \(m\geq 1\), then \(P\in \text{Ass}(R/(I^ n)^ a)\) for all \(n\geq m\) and, (b) if \(P\in \hat A^*(I)\), then \(P\in \hat A^*(IJ)\) for all ideals J of R such that height \((J)\geq 1\). (Here, if \(K\) is an ideal, then \(\hat A^*(K)=\{P\in \text{Spec}(R)\); \(P\in \text{Ass}(R/(K^ n)_ a)\) for some \(n\geq 1\)\}.)