Powers of ideals in locally unmixed Noetherian rings

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 (I n+i ) a C V for all i > 1, where (I n+ ) a is the integral closure of + . Moreover, if R is analytically unramified and / is any ideal in R, then (2) holds for / and, (3) There exists an integer m > 1 such that, with B = (I m ) a , (B i ) a = B for all i>\. The main result in this paper shows that an analogous theorem holds with reduced unmixed local ring and I [l] replacing analytically unramified semi-local ring and (/') , respectively, where / [/] is the intersection of certain primary ideals related to /'. An application and a generalization are included.