### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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.)

## Examples

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

$Spec R/I \to Spec R$

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

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

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

is called the de Rham space of $X$.

Last revised on October 26, 2017 at 15:42:01. See the history of this page for a list of all contributions to it.