\begin{definition}[Definition:Residue Division Ring Induced by Non-Archimedean Norm]
Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring.
Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$.
The '''residue division ring induced by the norm''' $\norm {\,\cdot\,}$ is the quotient ring $\OO / \PP$.
If $R$ is a field then the quotient ring $\OO / \PP$ is called the '''residue field induced by the norm''' $\norm {\,\cdot\,}$.
\end{definition}