Definition:Valuation

From ProofWiki
Jump to navigation Jump to search

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:

\((V1)\)   $:$     \(\displaystyle \forall a, b \in R:\)    \(\displaystyle \nu \left({a \times b}\right) \)   \(\displaystyle = \)   \(\displaystyle \nu \left({a}\right) + \nu \left({b}\right) \)             
\((V2)\)   $:$     \(\displaystyle \forall a \in R:\)    \(\displaystyle \nu \left({a}\right) = +\infty \)   \(\displaystyle \iff \)   \(\displaystyle a = 0_R \)             where $0_R$ is the ring zero
\((V3)\)   $:$     \(\displaystyle \forall a, b \in R:\)    \(\displaystyle \nu \left({a + b}\right) \)   \(\displaystyle \ge \)   \(\displaystyle \min \left\{ {\nu \left({a}\right), \nu \left({b}\right) }\right\} \)             


Also defined as

A valuation is usually defined on a field.

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