Definition 1

Let $C$ be a (not necessarily commutative) ring and $\Gamma$ a totally ordered commutative group written additively. A valuation on $C$ with values in $\Gamma$ is any mapping $v : C \to \Gamma_\infty$ which satisfies the following conditions:

(VL_I) $v(xy) = v(x) + v(y)$ for $x \in C, y \in C$. (VL_{II}) $v(x + y) \geq \inf(v(x), v(y))$ for $x \in C, y \in C$. (VL_{III}) $v(1) = 0$ and $v(0) = +\infty$.

If $C$ has no divisor of zero other than 0, the unique mapping $v_0$ of $C$ to $\Gamma_\infty$ such that $v_0(x) = 0$ for $x \neq 0$ and $v_0(0) = +\infty$ is a valuation, called the improper valuation on $C$. If $z \in C$ is such that $z^n = 1$ for some integer $n \geq 1$, then, by (VL_I), $nv(z) = v(z^n) = 0$ and hence $v(z) = 0$ for every valuation $v$ on $C$, since $\Gamma$ is a totally ordered group. In particular $v(-1) = 0$, whence $v(-x) = v(x)$ for all $x \in C$. Moreover, it follows from (VL_I) that $v(xy) = v(yx)$ for all $x, y$ in $C$. If $x$ is invertible in $C$, then $v(x^{-1}) = -v(x)$.