Integral closure of an affine algebra

Abstract

Let R be a commutative ring with identity and R ′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R ′ is finitely generated, i.e., R ′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R ≤ 2 , then R ′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R = ∞ or reg-dim R = n ≤ dim R = n + m − 1 for arbitrary positive integers n, m.