I was looking the valuation ring of dimension $2$. Then I found, it has two number of nonzero prime ideals and localization at prime is a valuation domain again. Moreover, there is a onetoone correspondence between prime ideals and valuation overrings. Dimension 2 means, we have two nonzero prime ideals, then the valuation ring should have two valuation overrings different from quotient field. We will get one by localize at prime ideal of height 1 and another one by localized at maximal ideal. I have difficulty to figure out the overrvaluation ring with respect to maximal ideal. That is, let $V$ be a valuation rings and $0\subseteq p\subseteq m$ then $V\subseteq V_{m} \subseteq V_{(p)}\subseteq V_{(0)} = K$, quotient field. The second sequence forces me that $V$ has dimension $3$ instead of $2$. Where is my mistake?
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