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

• $|-x|=|(-1)x|=|-1|\cdot |x|=|x|$