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.
$x=0\iff x=0$;

2.
$xy=x\cdot y$;

3.
$x+y\leqq \mathrm{max}\{x,y\}$.
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 $K\setminus \{0\}$ 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 nonarchimedean (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^{} ($S\cap {S}^{1}=\mathrm{\varnothing}$).

•
$x=(1)x=1\cdot x=x$
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  20130322 14:54:39 
Last modified on  20130322 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 