# The Stacks Project

## Tag 00JU

Lemma 10.56.8. Let $S$ be a graded ring.

1. Any minimal prime of $S$ is a homogeneous ideal of $S$.
2. Given a homogeneous ideal $I \subset S$ any minimal prime over $I$ is homogeneous.

Proof. The first assertion holds because the prime $\mathfrak q$ constructed in Lemma 10.56.7 satisfies $\mathfrak q \subset \mathfrak p$. The second because we may consider $S/I$ and apply the first part. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 13048–13056 (see updates for more information).

\begin{lemma}
Let $S$ be a graded ring.
\begin{enumerate}
\item Any minimal prime of $S$ is a homogeneous ideal of $S$.
\item Given a homogeneous ideal $I \subset S$ any minimal
prime over $I$ is homogeneous.
\end{enumerate}
\end{lemma}

\begin{proof}
The first assertion holds because the prime $\mathfrak q$ constructed in
Lemma \ref{lemma-smear-out} satisfies $\mathfrak q \subset \mathfrak p$.
The second because we may consider $S/I$ and apply the first part.
\end{proof}

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