Lemma 10.35.10. Let $R_ i\to S_ i$ be ring maps $i = 1, \ldots , n$. Denote the integral closure of $R_ i$ in $S_ i$ by $S'_ i$. Further let $R$ and $S$ denote the product of the $R_ i$ and $S_ i$ respectively. Then the integral closure of $R$ in $S$ is the product of the $S'_ i$. In particular $R \to S$ is integrally closed if and only if each $R_ i \to S_ i$ is integrally closed.

**Proof.**
This follows immediately from Lemma 10.35.8.
$\square$

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