nLab nilradical


Not to be confused with nilpotent ideal, which is a nilpotent element in the lattice of ideals of a ring.



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.)


Every local Artinian ring is a commutative ring whose set of non-invertible elements is a nilradical.

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.


See also:

Last revised on January 12, 2023 at 18:12:16. See the history of this page for a list of all contributions to it.