The integral closure of a Noetherian ring

Huckaba, James A.

doi:10.1090/S0002-9947-1976-0401734-6

The integral closure of a Noetherian ring
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Trans. Amer. Math. Soc. 220 (1976), 159-166 Request permission

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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