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:

Also defined as
A valuation is usually defined on a field.

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