# alternative definition of Krull valuation

Let $G$ be an abelian totally ordered group, denoted additively. We adjoin to $G$ a new element $\infty$ such that $g<+\infty$, for all $g\in G$ and we extend the addition on $G_{\infty}=G\cup\{+\infty\}$ by declaring $g+(+\infty)=(+\infty)+(+\infty)=+\infty$.

###### Definition 1.

Let $R$ be an unital ring, a valuation of $R$ with values in $G$ is a function from $R$ to $G_{\infty}$ such that , for all $x,y\in R$:

1) $v(xy)=v(x)+v(y)$,

2) $v(x+y)\geq\min\{v(x),v(y)\}$,

3) $v(x)=+\infty$ iff $v(x)=0$.

Remarks a) The condition 1) means that $v$ is a homomorhism of $R\smallsetminus\{0\}$ with multiplication in the group $G$. In particular, $v(1)=0$ and $v(-x)=v(x)$, for all $x\in G$. If $x$ is invertible then $0=v(1)=v(xx^{-1})=v(x)+v(x^{-1})$, so $v(x^{-1})=-v(x)$.
b) If 3) is replaced by the condition $v(0)=+\infty$ then the set $P=v^{-1}\{+\infty\}$ is a prime ideal of $R$ and $v$ is on the integral domain $R/P$.
c) In particular, conditions 1) and 3) that $R$ is an integral domain and let $K$ be its quotient field. There is a unique valuation of $K$ with values in $G$ that extends $v$, namely $v(x/y)=v(x)-v(y)$, for all $x\in R$ and $y\in R\smallsetminus\{0\}$.
d) The element $v(x)$ is sometimes denoted by $vx$.

 Title alternative definition of Krull valuation Canonical name AlternativeDefinitionOfKrullValuation Date of creation 2013-03-22 17:02:08 Last modified on 2013-03-22 17:02:08 Owner polarbear (3475) Last modified by polarbear (3475) Numerical id 10 Author polarbear (3475) Entry type Definition Classification msc 13F30 Classification msc 13A18 Classification msc 12J20 Classification msc 11R99 Related topic OrderValuation Related topic Valuation Related topic Krullvaluation Related topic KrullValuation