Let $R=\oplus_{i\geq 0}R_i$ be a graded ring. Denote $S$ as the ring $R$ without gradation structure. Suppose $p\in Spec(S)$. I want to consider $Q=\oplus_ip\cap R_i$. Suppose $ab\in Q\subset p$ with $a,b$ homogeneous. Since $p$ is a prime ideal of $S$ and $ab\in p\subset S$, WLOG, I can assume $a\in p$. From $a$ homogeneous, $a\in R_i$ for some $i$. Hence, I see $Q$ is prime.
$\textbf{Q:}$ Is above idea correct? It seems that every inhomogeneous ideal $p$ contains a "maximal" prime ideal $Q$ contained in it. I think the issue is $p\cap R_i$. This may need to generate a $R_0$ submodule of $R$ rather than naively taking intersection. If this is wrong, what is the counter example and what is the problem with above idea?