Lemma 10.50.9 (088Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.50: Valuation rings
Lemma 10.50.9 (cite)

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Lemma 10.50.9. Let $A$ be a valuation ring. For any prime ideal $\mathfrak p \subset A$ the quotient $A/\mathfrak p$ is a valuation ring. The same is true for the localization $A_\mathfrak p$ and in fact any localization of $A$ .

Proof. Use the characterization of valuation rings given in Lemma 10.50.5. $\square$

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