Definition:Valuation Axioms

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Definition

Let $\struct {R, +, \times}$ 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:

\((\text V 1)\)   $:$     \(\displaystyle \forall a, b \in R:\)    \(\displaystyle \map \nu {a \times b} \)   \(\displaystyle = \)   \(\displaystyle \map \nu a + \map \nu b \)             
\((\text V 2)\)   $:$     \(\displaystyle \forall a \in R:\)    \(\displaystyle \map \nu a = +\infty \)   \(\displaystyle \iff \)   \(\displaystyle a = 0_R \)             where $0_R$ is the ring zero
\((\text V 3)\)   $:$     \(\displaystyle \forall a, b \in R:\)    \(\displaystyle \map \nu {a + b} \)   \(\displaystyle \ge \)   \(\displaystyle \min \set {\map \nu a, \map \nu b} \)