Trivial Norm on Division Ring is Norm

Theorem
Let $\left({R, +, \circ}\right)$ be a division ring, and denote its ring zero by $0_R$.

Then the trivial norm $\left \Vert{\cdot}\right \Vert: R \to \R_{\ge 0}$, which is given by:


 * $\left\Vert{x}\right\Vert = \begin{cases}

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

defines a norm on $R$

Proof
We prove the norm axioms one by one:

This follows directly fom the definition of the trivial norm.

If $x=0_R$:

Similarly, if $y=0_R$.

If $x,y\ne 0_R$, then $x \circ y \ne 0_R$ by alternative definition $(3)$ of division rings. We get:

If $x=0_R$:

Similarly, if $y=0_R$.

If $x,y \ne 0_R$: