----- > [!proposition] Proposition. ([[a radical ideal in a Noetherian ring has only finitely many minimal prime ideals]]) > Let $R$ be a [[Noetherian ring]] and $I=\sqrt{ I } \subset R$ be a [[radical of an ideal|radical]] [[ideal]] of $R$. Then $I=\sqrt{ I }$ has only finitely many [[minimal prime ideal|minimal]] [[prime ideal|prime ideals]] over it. ^proposition > [!proof]- Proof. ([[a radical ideal in a Noetherian ring has only finitely many minimal prime ideals]]) > **Lemma.** *First recall that such an ideal is equal to the intersection of finitely many [[prime ideal|prime ideals]].* See [[a radical ideal in a Noetherian ring has only finitely many minimal prime ideals]]. > > **Proof of the statement.** Now let $I=\sqrt{ I } \subset R$ be a [[radical of an ideal|radical ideal]]; using the lemma write $I=\mathfrak{p}_{1} \cap\dots \cap \mathfrak{p}_{r}$ for some prime ideals $\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}$ of $R$. Let $\mathfrak{p}$ be a minimal prime ideal over $I$. By property in [[prime ideal]], $\mathfrak{p} \supset \mathfrak{p}_{i}=\mathfrak{q}$ for some $i$. But then $\mathfrak{q}=\mathfrak{p}_{i} \supset I$, and hence since $\mathfrak{p}$ is a minimal prime ideal the fact that $I \subset \mathfrak{q} \subset \mathfrak{p}$ implies that $\mathfrak{p}=\mathfrak{q}=\mathfrak{p}_{i}$ for some $i=1,\dots,r$. > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```