Definition:Valuation Ideal 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 ideal induced by the non-Archimedean norm $\norm{\,\cdot\,}$ is the set:

$\PP = \set{x \in R: \norm x \lt 1}$


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


Also see




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