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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. $x=0\,\,\Leftrightarrow\,\,x=0$;
2. $xy=x\cdoty$;
3. $x+y\leqq\max\{x,\,y\}$.
Thus the Krull valuation is more general than the usual valuation, which is also characterized as valuation of rank 1 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 means the rank of the value group.
We may say that a Krull valuation is nonarchimedean.
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 ($S\cap S^{{1}}=\varnothing$).

$x=(1)x=1\cdotx=x$
References
 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
 2 P. Jaffard: Les systèmes d’idéaux. Dunod, Paris (1960).
Mathematics Subject Classification
13F30 no label found13A18 no label found12J20 no label found11R99 no label found Forums
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