Lemma 10.37.10. Let $R$ be a domain. The following are equivalent:

The domain $R$ is a normal domain,

for every prime $\mathfrak p \subset R$ the local ring $R_{\mathfrak p}$ is a normal domain, and

for every maximal ideal $\mathfrak m$ the ring $R_{\mathfrak m}$ is a normal domain.

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