Lemma 10.35.14 (00G6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.35: Jacobson rings
Lemma 10.35.14 (cite)

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Lemma 10.35.14. Let $R$ be a Jacobson ring. Let $f \in R$ . The ring $R_ f$ is Jacobson and maximal ideals of $R_ f$ correspond to maximal ideals of $R$ not containing $f$ .

Proof. By Topology, Lemma 5.18.5 we see that $D(f) = \mathop{\mathrm{Spec}}(R_ f)$ is Jacobson and that closed points of $D(f)$ correspond to closed points in $\mathop{\mathrm{Spec}}(R)$ which happen to lie in $D(f)$ . Thus $R_ f$ is Jacobson by Lemma 10.35.4. $\square$

Comments (1)

Comment #1070 by Matthieu Romagny on October 12, 2014 at 08:35

I usually understand the word "correspond" to mean that there is a 1-1 correspondence. Thus I'd write "correspond to maximal ideals of $R$ not containing $f$ ".

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