---- > [!definition] Definition. ([[nilpotent ideal]]) > An [[ideal]] $I$ of a [[ring]] $R$ is said to be **nilpotent** if $I^{\ell}=\langle \{ x_{1} \cdots x_{\ell}: x_{i } \in I \} \rangle=(0)$ for some $\ell \geq 1$. ^definition > [!basicexample] > When $R$ is [[Noetherian ring|Noetherian]], the [[nilradical of a ring|nilradical]] $\text{Nil }R$ is a nilpotent ideal, per the corollary in [[every ideal in a Noetherian ring contains a power of its radical]]. ^basic-example ---- #### ---- #### 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 > ```