I'm trying to read a proof of the following proposition:
Let $S$ be a graded ring, $T \subseteq S$ a multiplicatively closed set. Then a homogeneous ideal maximal among the homogeneous ideals not meeting $T$ is prime.
In this proof, it says
"it suffices to show that if $a,b \in S$ are homogeneous and $ab \in \mathfrak{p}$, then either $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$"
where $\mathfrak{p}$ is our maximal homogeneous ideal. I don't know how to prove this is indeed sufficient. If I try writing general $a$ and $b$ in terms of "coordinates": $a=a_0+\cdots+a_n$ where $a_0 \in S_0,$ then I can see it working for small $n$, but it seems to get so complicated I wouldn't know how to write down a proof. Is there a better way to attack this problem?