Axiom:Multiplicative Norm Axioms

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

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

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