# Properties of Norm on Division Ring/Norm of Negative

## Theorem

Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.

Let $\norm{\,\cdot\,}$ be a norm on $R$.

Let $x \in R$

Then:

$\norm {-x} = \norm {x}$

## Proof

By Norm of Negative of Unity then:

$\norm{-1_R} = 1$.

Then:

 $\displaystyle \norm{-x}$ $=$ $\displaystyle \norm{-1_R \circ x}$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle \norm{-1_R} \norm{x}$ Norm axiom (N2) (Multiplicativity) $\displaystyle$ $=$ $\displaystyle \norm{x}$ Norm of Negative of Unity

as desired.

$\blacksquare$