Lemma 33.37.2. In Situation 33.37.1 assume that $B$ is a valuation ring. Then for every unit $u$ of $A$ either $u \in R$ or $u^{-1} \in R$.

Proof. Namely, if the image $c$ of $u$ in $K$ is in $B$, then $u \in R$. Otherwise, $c^{-1} \in B$ (Algebra, Lemma 10.50.3) and $u^{-1} \in R$. $\square$

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