Nilradical of a ring

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements:

${\displaystyle {\mathfrak {N}}_{R}=\lbrace f\in R\mid f^{m}=0{\text{ for some }}m\in \mathbb {Z} _{>0}\rbrace .}$

It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals.

In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this.

The nilradical of a Lie algebra is similarly defined for Lie algebras.