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

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 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 ($S\cap S^{-1}=\mathrm{\varnothing }$).

• $|-x|=|(-1)x|=|-1|\cdot |x|=|x|$

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