The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.35.4. 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 10.35.3. $\square$


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