Axiom:Non-Archimedean Norm Axioms

Non-Archimedean Norm Axioms (Division Ring)
Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$.

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

The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean norm:

Non-Archimedean Norm Axioms (Vector Space)
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $X$ be a vector space over $R$, with zero $0_X$.

Let $\norm {\, \cdot \,}: X \to \R_{\ge 0}$ be a mapping from $X$ to the non-negative reals.

$\norm {\, \cdot \,}$ is satisfies the non-Archimedean norm axioms $\norm {\, \cdot \,}$ satisfies the following contitions: