= Integral closure as an intersection of valuation rings
{title2=$\overline R^{K}=\bigcap_{R\subseteq V\subseteq K}V$}
For a domain $R$ with <fraction field> $K$, its <integral closure> is the intersection of all <valuation rings> of $K$ containing $R$. Integral elements belong to every such <ring> because <valuation rings are integrally closed>. For a nonintegral $x$, the <ideal> $x^{-1}R[x^{-1}]$ is proper, since its containing $1$ would give a monic equation for $x$. Localize at a <maximal ideal> containing $x^{-1}$ and apply the <valuation domination lemma>. The resulting <ring> contains $R$ but not $x$, proving the reverse inclusion.
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