Axiom:Valuation Axioms

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:

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