Lemma 10.57.7. Let $S$ be a graded ring. Let $\mathfrak p \subset S$ be a prime. Let $\mathfrak q$ be the homogeneous ideal of $S$ generated by the homogeneous elements of $\mathfrak p$. Then $\mathfrak q$ is a prime ideal of $S$.
Proof. To prove that $\mathfrak q$ is prime, it suffices to check that if $f, g \in S$ are homogeneous and $fg \in \mathfrak q$, then either $f$ or $g$ in $\mathfrak q$. Then $fg \in \mathfrak p$ because $\mathfrak p \subset \mathfrak q$. Since $\mathfrak p$ is prime we see that either $f \in \mathfrak p$ or $g \in \mathfrak p$. Since $f$ and $g$ are homogeneous, it then is clear that either $f \in \mathfrak q$ or $g \in \mathfrak q$. $\square$
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