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integrity characterized by places

\documentclass{article}
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\begin{document}
\begin{thmplain}
 \, Let $R$ be a subring of the field $K$, \,$1\in R$. \,An element $\alpha$ of the field is integral over $R$ if and only if all \PMlinkname{places}{PlaceOfField} $\varphi$ of $K$ satisfy the implication
     $$\varphi \mathrm{\,\,is\,finite\,in\,}R\,\,\,\Rightarrow
          \,\,\varphi(\alpha)\mathrm{\,is\,finite}.$$
\end{thmplain}

\textbf{\PMlinkescapetext{Corollary} 1.} \,Let $R$ be a subring of the field $K$, \,$1\in R$. \,The integral closure of $R$ in $K$ is the intersection of all valuation domains in $K$ which contain the ring $R$. \,The integral closure is integrally closed in the field $K$.


\textbf{\PMlinkescapetext{Corollary} 2.}  \,Every valuation domain is integrally closed in its field of fractions.
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