Definition:Valuation Ideal Induced by Non-Archimedean Norm

Definition
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with zero $0_R$.

The valuation ideal induced by the norm $\norm{\,\cdot\,}$ is the set:
 * $\mathcal O = \set{x \in R: \norm{x} \lt 1}$

That is, the valuation ideal induced by the norm $\norm{\,\cdot\,}$ is the Open ball $B_1 \paren {0_R}$.

Also See

 * Valuation Ideal is Maximal Ideal of Induced Valuation Ring - where it is shown that the valuation ideal is an ideal that is a maximal left ideal and a maximal right ideal of the valuation ring induced by the norm.


 * 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:Residue Division Ring Induced by Non-Archimedean Norm


 * Residue Division Ring is Non-Archimedean Normed Division Subring - where it is shown that the quotient ring of the valuation ring by the valuation ideal is a division ring.