nLab
nilradical

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Definition

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

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

(If RR 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

SpecR/ISpecR Spec R/I \to Spec R

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

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

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

is called the de Rham space of XX.

Revised on November 26, 2013 10:27:19 by Urs Schreiber (188.200.54.65)