Definition:Valuation Ring Induced by Non-Archimedean Norm
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Definition
Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$.
The valuation ring induced by the non-Archimedean norm $\norm{\,\cdot\,}$ is the set:
- $\OO = \set {x \in R: \norm x \le 1}$
That is, the valuation ring induced by the non-Archimedean norm $\norm{\,\cdot\,}$ is the closed ball $\map {B_1^-} {0_R}$.
Also See
- Valuation Ring of Non-Archimedean Division Ring is Subring: the valuation ring induced by the norm $\norm{\,\cdot\,}$ is a subring of $R$.
- Valuation Ideal is Maximal Ideal of Induced Valuation Ring: 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.
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Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.4$ Algebra: Definition $2.4.2$
- Weisstein, Eric W. "Valuation Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ValuationRing.html.html