Let $R=\oplus_{i\in\mathbb{Z}} R_i$ be a (commutative) graded ring of type $\mathbb{Z}$. It can be shown that if $S$ is a multiplicative set consists of homogeneous elements, $R_S$ have a natural grading structure of type $\mathbb{Z}$.
My question is:
If $\mathfrak{p}\in \mathrm{Spec}(R)$ (possibly not homogeneous), then is it true that $(R_S, \mathfrak{p}R_S)$ is a local ring, where $S=\{F\in R \mid F$ is homogeneous and $F\not\in \mathfrak{p}\}$?
I know that the subring (of degree $0$) $R_{S,0} \subset R_S$ is local, and if a graded ring $A$ is local then the subring (of degree $0$) $A_0 \subset A$ is local, too.
This is false. Let $R=k[x]$ for $k$ a field and let $\mathfrak{p}=(x)$. Then there are no nonconstant homogeneous elements not in $(x)$, so $S=k^\times$ and $R_S=k[x]$ which is not local.