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.