It is known that the
following statements are equivalent for a semi-local ring R: (1) R is analytically
unramified; (2) There exists an open ideal I in R and an integer n ≥ 0 such
that (In+i)a⊆ Ii for all i ≥ 1, where (In+i)a is the integral closure of In+i.
Moreover, if R is analytically unramified and I is any ideal in R, then (2) holds
for I and, (3) There exists an integer m ≥ 1 such that, with B = (Im)a,
(Bi)a= Bi for all i ≥ 1. The main result in this paper shows that an analogous
theorem holds with reduced unmixed local ring and I[i] replacing analytically
unramified semi-local ring and (Ii)a, respectively, where I[i] is the intersection of
certain primary ideals related to Ii. An application and a generalization are
included.
