# Definition:Non-Archimedean/Norm (Division Ring)/Definition 2

## Definition

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

A non-Archimedean norm on $R$ is a mapping from $R$ to the non-negative reals:

$\norm {\, \cdot \,}: R \to \R_{\ge 0}$

satisfying the non-Archimedean norm axioms:

 $(\text N 1)$ $:$ Positive Definiteness: $\displaystyle \forall x \in R:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = 0_R$ $(\text N 2)$ $:$ Multiplicativity: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x \circ y}$ $\displaystyle =$ $\displaystyle \norm x \times \norm y$ $(\text N 4)$ $:$ Ultrametric Inequality: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \max \set {\norm x, \norm y}$