## Tag `00JU`

Chapter 10: Commutative Algebra > Section 10.56: Proj of a graded ring

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

- Any minimal prime of $S$ is a homogeneous ideal of $S$.
- 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}
\label{lemma-graded-ring-minimal-prime}
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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