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The Stacks project

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

  1. The domain R is a normal domain,

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

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

Proof. We deduce (1) \Rightarrow (2) from Lemma 10.37.5. The implication (2) \Rightarrow (3) is immediate. The implication (3) \Rightarrow (1) follows from the fact that for any domain R we have

R = \bigcap \nolimits _{\mathfrak m} R_{\mathfrak m}

inside the fraction field of R. Namely, if g is an element of the right hand side then the ideal I = \{ x \in R \mid xg \in R\} is not contained in any maximal ideal \mathfrak m, whence I = R. \square


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