# definition of valuation ring of a place on wikipedia

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?

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.