Lemma 10.50.16. Let $(\Gamma , \geq )$ be a totally ordered abelian group. Let $K$ be a field. Let $v : K^* \to \Gamma $ be a homomorphism of abelian groups such that $v(a + b) \geq \min (v(a), v(b))$ for $a, b \in K$ with $a, b, a + b$ not zero. Then
\[ A = \{ x \in K \mid x = 0 \text{ or } v(x) \geq 0 \} \]
is a valuation ring with value group $\mathop{\mathrm{Im}}(v) \subset \Gamma $, with maximal ideal
\[ \mathfrak m = \{ x \in K \mid x = 0 \text{ or } v(x) > 0 \} \]
and with group of units
\[ A^* = \{ x \in K^* \mid v(x) = 0 \} . \]
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