Let $R$ be a commutative ring. An element $x\in R$ is said to be nilpotent if $x^{n}=0$ for some positive integer $n$. The set of all nilpotent elements of $R$ is an ideal of $R$, called the nilradical of $R$ and denoted $\operatorname{Nil}(R)$. The nilradical is so named because it is the radical of the zero ideal.
The nilradical of $R$ equals the prime radical of $R$, although proving that the two are equivalent requires the axiom of choice.