Definition 2.2. Let $(K, | ∙ |)$ be a non-archimedean valued field. The valuation ring of $K$ is defined to be

\begin{array}{l} O_{K} & = {x \in K | | x | \leq 1} (= \bar{B} (0, 1)) \\ ( & = {x \in K^{\times} | v (x) \geq 0} \cup {0}) \end{array}