Valuation on a Dedekind domain

Question

If we take $R$ a Dedekind domain and $p$ a non zero prime ideal of $R$, we denote by $v_p$ the valuation of its quotient field $K$ with valuation ring $R_p=(R\setminus p)^{-1}R$. How can I show that for almost $p$, for $a \in K^*$ we have $v_p(a)=0$ ?

Answer 1

You mean that given $a\in K^*$ then $v_p(a)=0$ for all but finitely many prime ideals $p$.

Write $a=b/c$ where $b$, $c$ are nonzero elements of $R$, and observe that $b$ and $c$ lie in only finitely many prime ideals of $R$.