Lemma 10.50.14. Let $A$ be a valuation ring. The valuation $v : A -\{ 0\} \to \Gamma _{\geq 0}$ has the following properties:

$v(a) = 0 \Leftrightarrow a \in A^*$,

$v(ab) = v(a) + v(b)$,

$v(a + b) \geq \min (v(a), v(b))$ provided $a + b \not= 0$.

Lemma 10.50.14. Let $A$ be a valuation ring. The valuation $v : A -\{ 0\} \to \Gamma _{\geq 0}$ has the following properties:

$v(a) = 0 \Leftrightarrow a \in A^*$,

$v(ab) = v(a) + v(b)$,

$v(a + b) \geq \min (v(a), v(b))$ provided $a + b \not= 0$.

**Proof.**
Omitted.
$\square$

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