Axiom:Multiplicative Norm Axioms

Definition
Let $\left({R, +, \times}\right)$ be a division ring whose zero is denoted $0_R$.

Let $\left\Vert{\cdot}\right\Vert: R \to \R_{\ge 0}$ be a norm on $R$.

The norm axioms are the following conditions on $\left\Vert{\cdot}\right\Vert$ which define $\left\Vert{\cdot}\right\Vert$ as being a norm: