Lemma 10.57.8. 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.

Lemma 10.57.8. 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 10.57.7 satisfies $\mathfrak q \subset \mathfrak p$. The second because we may consider $S/I$ and apply the first part.
$\square$

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