---- > [!definition] Definition. ([[nilradical of a ring]]) > The **nilradical** of a [[commutative ring|commutative]] [[ring]] $R$ is the [[ideal]] consisting of [[nilpotent element of a ring|nilpotents]] $\text{Nil}_{R}=\{ f \in R : f^{n} =0 \text{ for some }n \in \mathbb{N} \}.$ It is thus the [[radical of an ideal|radical]] of the zero ideal. ^definition > [!equivalence] > - [[nilradical equals intersection of all prime ideals]] ^equivalence ---- #### ---- #### 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 > ```