Definition:Trivial Norm/Division Ring

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Let $\struct {R, +, \circ}$ be a division ring, and denote its zero by $0_R$.

Then the map $\norm {\cdot}: R \to \R_{\ge 0}$ given by:

$\norm x = \begin{cases} 0 & : \text{if $x = 0_R$}\\ 1 & : \text{otherwise} \end{cases}$

defines a norm on $R$, called the trivial norm.

Also known as

Some authors refer to this norm as the trivial absolute value.


A norm $\norm {\, \cdot \,}$ on a division ring $R$ is nontrivial if and only if it is not trivial.

Also see