Definition:Non-Archimedean/Norm (Division Ring)/Definition 1

Definition

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

A norm $\norm {\, \cdot \,}$ on $R$ is non-Archimedean if and only if $\norm {\, \cdot \,}$ satisfies the axiom:

 $(\text N 4)$ $:$ Ultrametric Inequality: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \max \set {\norm x, \norm y}$