1. Integral closure of an affine algebra.
- Author
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Chang, Gyu Whan and Kang, Byung Gyun
- Subjects
- *
PRIME ideals , *RING theory , *COMMUTATIVE rings , *ALGEBRA , *INTEGERS - Abstract
AbstractLet
R be a commutative ring with identity and R′ be the integral closure ofR . In this paper, we show that ifR is an affine algebra over a fieldK , 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 ifR is a Krull ring such thatR /P is a Noetherian domain for each minimal regular prime idealP ofR , thenR is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that ifR 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 ringR that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integersn ,m . [ABSTRACT FROM AUTHOR]- Published
- 2024
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