Definition:Valuation Axioms

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Definition

Let $\left({R, +, \times}\right)$ be a ring.

Let $\nu$ be a valuation on $R$.

The valuation axioms are the following conditions on $\nu$ which define $\nu$ as being a valuation:

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