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The Stacks project

Proof. It suffices to show that the product of two normal rings, say R and S, is normal. By Lemma 10.21.3 the prime ideals of R\times S are of the form \mathfrak {p}\times S and R\times \mathfrak {q}, where \mathfrak {p} and \mathfrak {q} are primes of R and S respectively. Localization yields (R\times S)_{\mathfrak {p}\times S}=R_{\mathfrak {p}} which is a normal domain by assumption. Similarly for S. \square


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