Prime divisors and divisorial ideals

Abstract

Let I 1,…, I g be regular ideals in a Noetherian ring R. Then it is shown that there exist positive integers k 1,…, k g such that ( I 1 n 1 + m 1 … I g n g + m g ):( I 1 m 1 … I g m g ) = I 1 n 1 … I g n g for all n i ≥ k i ( i = 1,…, g) and for all nonnegative integers m 1,..., m g . Using this, it is shown that if Δ is a multiplicatively closed set of nonzero ideals of R that satisfies certain hypotheses, then the sets Ass ( R (I 1 n 1 …I g n g ) ) equal for all large positive integers n 1,..., n g . Also, if R is locally analytically unramified, then some related results for general sets Δ are proved.