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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The integral closure of a Noetherian ring


Author: James A. Huckaba
Journal: Trans. Amer. Math. Soc. 220 (1976), 159-166
MSC: Primary 13B20
DOI: https://doi.org/10.1090/S0002-9947-1976-0401734-6
MathSciNet review: 0401734
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Abstract: Let R be a commutative ring with identity and let $ R'$ denote the integral closure of R in its total quotient ring. The basic question that this paper is concerned with is: What finiteness conditions does the integral closure of a Noetherian ring R possess? Unlike the integral domain case, it is possible to construct a Noetherian ring R of any positive Krull dimension such that $ R'$ is non-Noetherian. It is shown that if $ \dim R \leqslant 2$, then every regular ideal of $ R'$ is finitely generated. This generalizes the situation that occurs in the integral domain case. In particular, it generalizes Nagata's Theorem for two-dimensional Noetherian domains.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0401734-6
Keywords: Integral closure, Noetherian ring, valuation ring, Krull dimension of a ring
Article copyright: © Copyright 1976 American Mathematical Society

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