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

$A^* = \{ x \in K^* \mid v(x) = 0 \} .$

Proof. Omitted. $\square$

Comment #786 by Wei Xu on

The homomorphism $v$ should also have the property that $v(a+b)\geq \min(v(a),v(b)$ for $a,b,a+b \in K^{*}$.

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