Lemma 10.161.7 (032K)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.161: Japanese rings
Lemma 10.161.7 (cite)

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Lemma 10.161.7. Let $R$ be a Noetherian domain, and let $R \subset S$ be a finite extension of domains. If $S$ is N-1, then so is $R$ . If $S$ is N-2, then so is $R$ .

Proof. Omitted. (Hint: Integral closures of $R$ in extension fields are contained in integral closures of $S$ in extension fields.) $\square$

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