Nilradical

Let $R$ be a commutative ring. The nilradical of $R$ is the set

i.e. the set of all nilpotent elements of $R$; it is an ideal (indeed, the radical of the zero ideal in the sense of radicals of ideals ).

The nilradical can also be characterized as the intersection of all prime ideals (a standard fact recorded as nilradical = intersection of primes ). A ring is reduced precisely when its nilradical is zero.

Examples:

• If $R$ is reduced, then its nilradical is $\{0\}$.
• In $k[x]/(x^n)$, the nilradical is the ideal generated by the class of $x$.
• In $\mathbb Z/n\mathbb Z$, the nilradical consists of classes divisible by every prime dividing $n$.