Residual division, semi-prime operations, and a theorem of Sakuma and Okuyama

Abstract

Let I be a regular ideal of a Noetherian ring R. Then it is well known that: (a) I n+ k : I n = I k for all large k and for all n ⩾ 0; (b) if I is principal and H is another ideal of R, then I n+ j H m : I j = I n H m = I n ( I j H m : I j ) for all m⩾0, j⩾0, and n⩾1; and (c) if R is local and analytically unramified, then ( I n+ k ) a = I n ( I k ) a for all large k and for all n⩾0, where ( I j ) a is the integral closure of I j . The main results in this paper generalize these three theorems to the case where H and I are finite collections of Noetherian filtrations on R, and these new results are then used to show that a semi-local ring R is analytically unramified if and only if for every regular ideal I of R there exists a regular ideal K of R such that ( I n ) a = I n K : K for all n⩾1.