Lemma 10.36.4. Let $\varphi : R \to S$ be a ring map. Let $s_1, \ldots , s_ n$ be a finite set of elements of $S$. In this case $s_ i$ is integral over $R$ for all $i = 1, \ldots , n$ if and only if there exists an $R$-subalgebra $S' \subset S$ finite over $R$ containing all of the $s_ i$.

Proof. If each $s_ i$ is integral, then the subalgebra generated by $\varphi (R)$ and the $s_ i$ is finite over $R$. Namely, if $s_ i$ satisfies a monic equation of degree $d_ i$ over $R$, then this subalgebra is generated as an $R$-module by the elements $s_1^{e_1} \ldots s_ n^{e_ n}$ with $0 \leq e_ i \leq d_ i - 1$. Conversely, suppose given a finite $R$-subalgebra $S'$ containing all the $s_ i$. Then all of the $s_ i$ are integral by Lemma 10.36.3. $\square$

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