---- > [!definition] Definition. ([[minimal prime ideal]]) > For an [[ideal]] $I$ of a [[commutative ring|(commutative)]] [[ring]] $R$, a **minimal prime ideal over $I$** is a [[prime ideal]] $\mathfrak{p}$ of $R$ such that $I \subset \mathfrak{p}$ and if $I \subset \mathfrak{q} \subset \mathfrak{p}$ for some [[prime ideal]] $\mathfrak{q}$ then $\mathfrak{q}=\mathfrak{p}$. > A **minimal prime ideal of $R$** is defined to be a minimal prime ideal over $(0)$, or, equivalently, a minimal prime ideal over $\sqrt{ (0) }$ [[nilradical of a ring|since]] every [[prime ideal]] of $R$ contains $\sqrt{ (0) }$. ^definition ---- #### ---- #### 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 > ```