Lemma 15.124.7. The localization of a Bézout domain is Bézout. Every local ring of a Bézout domain is a valuation ring. A local domain is Bézout if and only if it is a valuation ring.
Proof. We omit the proof of the statement on localizations. The final statement is Algebra, Lemma 10.50.15. The second statement follows from the other two. $\square$
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