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 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 is non-Noetherian. It is shown that if , then every regular ideal of 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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Additional Information

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