Nilradical

Remarks

$N(R)$ is the radical $\sqrt{(0)}$ of the zero ideal, so it is an ideal of $R$. Example: in $\Z/12\Z$, $N = \set{0, 6}$; indeed $6^2 = 0$, and if $r^n \equiv 0 \pmod{12}$ then $2$ and $3$ divide $r$. Non-example: $2 \notin N(\Z/12\Z)$, because $2^n$ is never divisible by $3$. The quotient $R/N(R)$ has zero nilradical, that is, it is reduced. $N(R)$ is the intersection of all prime ideals of $R$; see the nilradical as an intersection of primes. Other notations are $\operatorname{nil}(R)$ and $\mathfrak{N}_R$.