Properties of Norm on Division Ring/Norm of Unity

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

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

Then:
 * $\norm {1_R} = 1$.

Proof
By :
 * $\forall x, y \in R: \norm {x \circ y} = \norm x \norm y$

In particular:
 * $\norm {1_R} = \norm {1_R \circ 1_R} = \norm {1_R} \norm {1_R}$

By :
 * $\norm {1_R} \ne 0$

So $\norm {1_R}$ has an inverse in $R$.

Multiplying by this inverse:
 * $\norm {1_R} \norm {1_R} = \norm {1_R} \iff \norm {1_R} = 1$

as desired.