# Krull valuation

Definition.  The mapping  $|.|\!:\,K\to G$,  where $K$ is a field and $G$ an ordered group equipped with zero, is a Krull valuation of $K$, if it has the properties

1. 1.

$|x|=0\,\,\Leftrightarrow\,\,x=0$;

2. 2.

$|xy|=|x|\cdot|y|$;

3. 3.

$|x+y|\leqq\max\{|x|,\,|y|\}$.

Thus the Krull valuation is more general than the usual valuation (http://planetmath.org/Valuation), which is also characterized as and which has real values.  The image  $|K\!\smallsetminus\!\{0\}|$  is called the value group of the Krull valuation; it is abelian.  In general, the rank of Krull valuation the rank (http://planetmath.org/IsolatedSubgroup) of the value group.

We may say that a Krull valuation is non-archimedean (http://planetmath.org/Valuation).

## Some values

• $|1|=1$   because the Krull valuation is a group homomorphism from the multiplicative group of $K$ to the ordered group.

• $|-1|=1$   because  $1=|(-1)^{2}|=|-1|^{2}$   and 1 is the only element of the ordered group being its own inverse ($S\cap S^{-1}=\varnothing$).

• $|-x|=|(-1)x|=|-1|\cdot|x|=|x|$

## References

• 1 Emil Artin: Theory of Algebraic Numbers.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
• 2 P. Jaffard: Les systèmes d’idéaux.  Dunod, Paris (1960).
 Title Krull valuation Canonical name KrullValuation Date of creation 2013-03-22 14:54:39 Last modified on 2013-03-22 14:54:39 Owner pahio (2872) Last modified by pahio (2872) Numerical id 19 Author pahio (2872) Entry type Definition Classification msc 13F30 Classification msc 13A18 Classification msc 12J20 Classification msc 11R99 Related topic OrderedGroup Related topic TrivialValuation Related topic IsolatedSubgroup Related topic ValueGroupOfCompletion Related topic PlaceOfField Related topic OrderValuation Related topic AlternativeDefinitionOfValuation2 Related topic UniquenessOfDivisionAlgorithmInEuclideanDomain Defines value group Defines rank of Krull valuation Defines rank of valuation