Let $R$ be a domain which is noetherian and a finitely generated $k$-algebra, i.e. $R\cong k[x_{1},...,x_{n}]/I$ for some ideal $I$ such that $\operatorname{Spec}(R)$ has Krull dimension $1$.
I want to prove that if we pick an element $f\in\operatorname{Frac}(R)$ that it is not contained in the localization of $R$ at the maximal ideal $\mathfrak{m}$ for only finitely many maximal ideals $\mathfrak{m}$.
I am not sure if this statement is true so a counterexample would also be nice.
Note that if we pick $f\in\operatorname{Frac}(R)$, then $f=g/h$ for some $g\in R$ and $h\in R\backslash\{0\}$. Notice that $f\in R_{\mathfrak{m}}$ if and only if there exists a $p\in R$ and $q,s\in R\backslash\mathfrak{m}$ such that $(gq-hp)s=0$. Since $R$ is a domain this is true if and only if $gq=hp$.
Added: As mentioned in the comments, the assumption that $R$ is a finitely generated $k$-algebra is unnecessary.