Lemma 10.37.14 (00H1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.37: Normal rings
Lemma 10.37.14 (cite)

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Lemma 10.37.14. Let $R$ be a normal ring. Then $R[x]$ is a normal ring.

Proof. Let $\mathfrak q$ be a prime of $R[x]$ . Set $\mathfrak p = R \cap \mathfrak q$ . Then we see that $R_{\mathfrak p}[x]$ is a normal domain by Lemma 10.37.8. Hence $(R[x])_{\mathfrak q}$ is a normal domain by Lemma 10.37.5. $\square$

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