Not to be confused with nilpotent ideal, which is a nilpotent element in the lattice of ideals of a ring.
symmetric monoidal (∞,1)-category of spectra
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.)
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
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
is called the de Rham space of $X$.
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.