= Valuation domination lemma
{title2=$A\subseteq V,\quad\mathfrak m_V\cap A=\mathfrak m_A$}
Every local subring $A$ of a <field> $K$ is dominated by a <valuation ring> with <fraction field> $K$. Order local overrings by inclusion and contraction of their maximal <ideals>; chain unions are local, so <Zorn's lemma> gives a maximal one. For each $z\in K^\times$, at least one of the extensions of its <maximal ideal> to $V[z]$ and $V[z^{-1}]$ is proper: otherwise two least-degree relations for $1$ with maximal-ideal coefficients reduce one another to a smaller-degree relation. Localizing the proper extension gives a dominating overring, so maximality forces either $z$ or $z^{-1}$ into $V$.
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