Nilradical equals intersection of prime ideals

Nilradical equals intersection of prime ideals: Let $R$ be a commutative ring. Then

i.e., an element is nilpotent if and only if it lies in every prime ideal. Equivalently, $\mathrm{Nil}(R)=\sqrt{(0)}$.

The nilradical is the ideal of all nilpotent elements ; this proposition expresses it as an intersection of prime ideals . Equivalently, it is the radical of $(0)$, so $R/\mathrm{Nil}(R)$ is reduced .