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.