Definition:Residue Division Ring Induced by Non-Archimedean Norm

Definition
Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring.

Let $\mathcal O$ be the valuation ring induced by the non-Archimedean norm $\norm{\,\cdot\,}$.

Let $\mathcal P$ 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 $\mathcal O / \mathcal P$.

If $R$ is a field then the quotient ring $\mathcal O / \mathcal P$ 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 $\mathcal O / \mathcal P$ is a division ring.