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In the page Valuation in Wikipedia, https://en.wikipedia.org/wiki/Valuation_(algebra) in the section "associated objects" it is written that we can associate to a valuation $v: K\to \mathbb{R}\cup \{\infty\} $its valuation ring $$R_v = \{x \in K \ |\ v(x) \geq 0 \}$$ the problem is that this object is only multiplicatively closed so it's not a ring. For example with $K = \mathbb{Q}$ and $v (x) = -\log|x|$ ($v(0) = \infty)$ we get that $\mathbb{Q}_v = \{|x|\leq 1\}$ which is not closed under addition.

Am I missing something?

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Yes, you are missing the last property, that $ v(a + b) \geq \operatorname{min}(v(a), v(b)) $ with equality whenever $ v(a) \not = v(b) $. Take $ a, b = 1 $ for instance.

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  • $\begingroup$ Oh I see, you are suggesting that the def. of valuation ring I am considering makes sense only for non-Archimedean absolute values. $\endgroup$ – Warlock of Firetop Mountain Jan 7 at 16:15

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