nLab nilradical

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

---

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

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

$$Spec R/I \to Spec R$$

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 $X$.

Related concepts

• infinitesimal extension

• reduced ring

• Artinian local ring

References

See also:

• Wikipedia, Nilradical