Lemma 10.50.14 (00IF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.50: Valuation rings
Lemma 10.50.14 (cite)

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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. $\square$

Comments (4)

Comment #43 by Rankeya on August 30, 2012 at 00:10

valution should be valuation

Comment #49 by Johan on August 30, 2012 at 11:37

Fixed. Thanks!

Comment #8871 by Zhenhua Wu on March 22, 2024 at 23:23

$a+b$ could happen to be $0$ which is out of the domain of definition. We could replace $\Gamma$ by $\Gamma\cup \{\infty\}$ and set $v(0)=\infty$ to avoid that. Actually the three properties also work in $v:K\to \Gamma\cup \{\infty\}$ .

Comment #9215 by Stacks project on May 15, 2024 at 15:42

Thanks and fixed here.

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