Lemma 10.35.8. Let $R_ i\to S_ i$ be ring maps $i = 1, \ldots , n$. Let $R$ and $S$ denote the product of the $R_ i$ and $S_ i$ respectively. Then an element $s = (s_1, \ldots , s_ n) \in S$ is integral over $R$ if and only if each $s_ i$ is integral over $R_ i$.

**Proof.**
Omitted.
$\square$

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