alternative definition of Krull valuation

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

Definition 1.

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

1) $v\mathit{}\mathrm{(}x\mathit{}y\mathrm{)}\mathrm{=}v\mathit{}\mathrm{(}x\mathrm{)}\mathrm{+}v\mathit{}\mathrm{(}y\mathrm{)}$,

2) $v\mathit{}\mathrm{(}x\mathrm{+}y\mathrm{)}\mathrm{\ge }\min \mathit{}\mathrm{\{}v\mathit{}\mathrm{(}x\mathrm{)}\mathrm{,}v\mathit{}\mathrm{(}y\mathrm{)}\mathrm{\}}$,

3) $v\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\mathrm{+}\mathrm{\infty }$ iff $v\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\mathrm{0}$.

Remarks a) The condition 1) means that $v$ is a homomorhism of $R\setminus \{0\}$ 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 $0=v(1)=v(x{x}^{-1})=v(x)+v(x^{-1})$, so $v(x^{-1})=-v(x)$.
b) If 3) is replaced by the condition $v(0)=+\mathrm{\infty }$ then the set $P=v^{-1}\{+\mathrm{\infty }\}$ 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 $y\in R\setminus \{0\}$.
d) The element $v(x)$ is sometimes denoted by $vx$.