# Definition:Valuation

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## 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**:

\((\text V 1)\) | $:$ | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle \map \nu {a \times b} \) | \(\displaystyle = \) | \(\displaystyle \map \nu a + \map \nu b \) | |||

\((\text V 2)\) | $:$ | \(\displaystyle \forall a \in R:\) | \(\displaystyle \map \nu a = +\infty \) | \(\displaystyle \iff \) | \(\displaystyle a = 0_R \) | where $0_R$ is the ring zero | ||

\((\text V 3)\) | $:$ | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle \map \nu {a + b} \) | \(\displaystyle \ge \) | \(\displaystyle \min \set {\map \nu a, \map \nu b} \) |

## Also defined as

A **valuation** is usually defined on a field.

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