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

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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:

\((N4)\)   $:$   Ultrametric Inequality:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \max \set {\norm x, \norm y} \)             


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