The Stacks project

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

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

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

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

Proof. Omitted. $\square$

Comments (4)

Comment #43 by Rankeya on

valution should be valuation

Comment #8871 by Zhenhua Wu on

could happen to be which is out of the domain of definition. We could replace by and set to avoid that. Actually the three properties also work in .

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  • 3 comment(s) on Section 10.50: Valuation rings

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