Definition:Non-Archimedean/Norm (Division Ring)

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

Definition 2
The pair $\struct {R, \norm {\, \cdot \, } }$ is a non-Archimedean Normed Division Ring.

If $R$ is also a commutative ring, that is, $\struct {R, \norm {\,\cdot\,} }$ is a valued field, then $\struct {R, \norm {\,\cdot\,} }$ is a non-Archimedean Valued Field.

Also see

 * Equivalence of Definitions of Non-Archimedean Division Ring Norm