nLab Artinian local ring

Definition

A Artinian local ring (or similar, also called a Weil ring) is a local ring which is also an Artinian ring; it satisfies the descending chain condition.

Crucially, this implies that the Jacobson radical of the local ring is a nilradical.

When an Artinian local ring is equipped with the structure of an algebra over a field (such as its residue field when the ring has equal characteristic) it is also referred to as an Artinian local algebra (or similar) or, especially in synthetic differential geometry, as a Weil algebra (in the sense of functions on an infinitesimally thickened point, but beware that the last term is ambiguous, see there).

Properties

General

Every Artinian local $\mathbb{K}$-algebra $A$ has a maximal ideal $\mathfrak{m}_A$, whose residue field $A / \mathfrak{m}_A$ is $\mathbb{K}$ itself. As a $\mathbb{K}$ vector space one has a splitting $A=\mathbb{K}\oplus \mathfrak{m}_A$. Moreover, the descending chain condition implies that $(\mathfrak{m}_A)^n=0$ for some $n\gg 0$, a consequence of Nakayama lemma. This implies that the maximal ideal is a nilradical.

Given a field $K$ and a Artinian local $K$-algebra $A$, let $I$ be the nilradical of $A$. There is a function $v:I \to \mathbb{N}$ which takes a nilpotent element $r \in I$ to the least natural number $n$ such that $r^{v(r) + 1} = 0$, such that given nilpotent elements $r \in I$ and $s \in I$ and non-zero scalars $a \in K$ and $b \in K$,

• $v(a r + b s) = v(r) + v(s)$

• $v(r s) = \min(v(r), v(s))$

In classical mathematics, Artinian local rings are the zero-dimensional local rings, in that every element is either invertible or nilpotent. However, in constructive mathematics, it is only the residually discrete Artinian local rings which are zero-dimensional local rings.

Spectra

Passing from commutative rings to their spectra (in the sense of algebraic geometry), Artinian local algebras correspond to infinitesimal pointed spaces. As such, they appear as bases of deformations in infinitesimal deformation theory. For instance $Spec(\mathbb{K}[\epsilon]/(\epsilon^2))$ is the base space for 1-dimensional first order deformations. Similarly, $Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1}))$ is the base space for 1-dimensional $n$-th order deformations.

An Artinian local algebra has a unique prime ideal, which means that its spectrum consists of a single point, i.e., $Spec(A)$ is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is $A$. By this reason spectra of Artinian local algebras are occasionally called fat points in the literature.

Examples

Example

The ring of dual numbers $\mathbb{K}[\epsilon]/(\epsilon^2)$ over a field $\mathbb{K}$.

Example

Every prime power local ring is a local Artinian ring.

Beware that some of these (for instance $\mathbb{Z}_{/4}$) are not of equal characteristic, and hence do not form algebras over their residue field.

See also

commutative ring	reduced ring	integral domain
local ring	reduced local ring	local integral domain
Artinian ring	semisimple ring	field
Weil ring	field	field

References

• The Stacks Project

  • 10.53. Artinian rings

  • 10.100. Flatness criteria over Artinian rings

Local Artinian $\infty$-algebras are discussed in

• Jacob Lurie, Formal moduli problems (pdf)