nLab
nilradical

Definition

For $R$ a commutative ring, its nilradical $I\subset R$ is the ideal of nilpotent elements: the collection of those elements $a\in R$ such that there is $n\in \mathbb{N}$ with $a^n=0$.

The quotient $R/I$ is also called the reduced part of $R$.

(If $R$ is not commutative there are different generalization of the notion of nilradical. See wikipedia, for the moment.)

Related concepts

With rings regarded as formal duals of affine schemes, the canonical inclusion

$$\mathrm{Spec}R/I\to \mathrm{Spec}R$$Spec R/I \to Spec R

is to be thought of as exhibiting the inclusion of $\mathrm{Spec}R/I$ into an infinitesimal thickening of itself.

For $X:\mathrm{CRing}\to \mathrm{Set}$ a presheaf on the category of commutative rings, the presheaf

$$X_{\mathrm{dR}}:\mathrm{Spec}R\mapsto X(\mathrm{Spec}R/I)$$X_{dR} : Spec R \mapsto X(Spec R/I)

is called the de Rham space of $X$.