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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