Definition:Residue Division Ring Induced by Non-Archimedean Norm

Definition
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 $\norm {\,\cdot\,}$.

Also see

 * Definition:Valuation Ring Induced by Non-Archimedean Norm


 * Valuation Ring of Non-Archimedean Division Ring is Subring, where it is shown that the valuation ring induced by the norm $\norm {\,\cdot\,}$ is a subring of $R$.


 * Definition:Valuation Ideal Induced by Non-Archimedean Norm


 * Valuation Ideal is Maximal Ideal of Induced Valuation Ring, where it is shown that the valuation ideal is an ideal such that the quotient ring $\OO / \PP$ is a division ring.