# Definition:Valuation

## Definition

Let $\left({R, +, \cdot}\right)$ be a ring.

A valuation on $R$ is a mapping:

$\nu: R \to \Z \cup \left\{{+\infty}\right\}$

which fulfils the valuation axioms:

 $(\text V 1)$ $:$ $\ds \forall a, b \in R:$ $\ds \map \nu {a \times b}$ $\ds =$ $\ds \map \nu a + \map \nu b$ $(\text V 2)$ $:$ $\ds \forall a \in R:$ $\ds \map \nu a = +\infty$ $\ds \iff$ $\ds a = 0_R$ where $0_R$ is the ring zero $(\text V 3)$ $:$ $\ds \forall a, b \in R:$ $\ds \map \nu {a + b}$ $\ds \ge$ $\ds \min \set {\map \nu a, \map \nu b}$

## Also defined as

A valuation is usually defined on a field.

However, the valuation axioms are as equally well defined on a ring.