# Definition:Valuation Axioms

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 $(V3)$ $:$ $\displaystyle \forall a, b \in R:$ $\displaystyle \nu \left({a + b}\right)$ $\displaystyle \ge$ $\displaystyle \min \left\{ {\nu \left({a}\right), \nu \left({b}\right) }\right\}$