Vol. 107, No. 2, 1983

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Powers of ideals in locally unmixed Noetherian rings

Louis Jackson Ratliff, Jr.

Vol. 107 (1983), No. 2, 459–472

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.

Mathematical Subject Classification 2000
Primary: 13C15
Secondary: 13A15, 13J10
Received: 18 September 1981
Revised: 4 June 1982
Published: 1 August 1983
Louis Jackson Ratliff, Jr.