On asymptotic prime divisors

Louis Jackson Ratliff, Jr.

Vol. 111 (1984), No. 2, 395–413

DOI: 10.2140/pjm.1984.111.395

Abstract

Several results are proved concerning the set Â^∗(I) = {P ∈ Spec R; P is a prime divisor of the integral closure (Iⁱ)_a of Iⁱ for all large i}, where I is an ideal in a Noetherian ring R. Among these are: if P is a prime divisor of (Iⁱ)_a for some i ≥ 1, then P is a prime divisor of (Iⁿ)_a for all n ≥ i; a characterization of Cohen-Macaulay rings and of altitude two local UFDs in terms of Â^∗(I); and, some results on the relationship of Â^∗(I) to Â^∗(IS) with S a flat R-algebra and to Â^∗((I + z)∕z) with z a minimal prime ideal in R.

Mathematical Subject Classification 2000

Primary: 13A15

Secondary: 13E05, 13B20

Milestones

Received: 11 June 1982

Revised: 10 August 1982

Published: 1 April 1984