nLab Artinian local ring




Formal geometry



A Artinian local ring or local Artinian ring or local Artin ring or Artin local ring or Weil ring is a local ring which satisfies one of the equivalent conditions

These are also called Artinian local algebras, local Artinian algebras, Artin local algebras, or local Artin algebras or Weil algebras, since Artinian local rings are commutative algebras over the residue field 𝕂\mathbb{K}.

In synthetic differential geometry, the term Weil algebra is used for real Artinian local algebras, see infinitesimally thickened point for more details.


Every Artinian local 𝕂\mathbb{K}-algebra AA has a maximal ideal 𝔪 A\mathfrak{m}_A, whose residue field A/𝔪 AA / \mathfrak{m}_A is 𝕂\mathbb{K} itself. As a 𝕂\mathbb{K} vector space one has a splitting A=𝕂𝔪 AA=\mathbb{K}\oplus \mathfrak{m}_A. Moreover, the descending chain condition implies that (𝔪 A) n=0(\mathfrak{m}_A)^n=0 for some n0n\gg 0, a consequence of Nakayama lemma. This implies that the maximal ideal is a nilradical.

Given a field KK and a Artinian local KK-algebra AA, let II be the nilradical of AA. There is a function v:Iv:I \to \mathbb{N} which takes a nilpotent element rIr \in I to the least natural number nn such that r v(r)+1=0r^{v(r) + 1} = 0, such that given nilpotent elements rIr \in I and sIs \in I and non-zero scalars aKa \in K and bKb \in K,

  • v(ar+bs)=v(r)+v(s)v(a r + b s) = v(r) + v(s)
  • v(rs)=min(v(r),v(s))v(r s) = \min(v(r), v(s))


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(𝕂[ϵ]/(ϵ 2))Spec(\mathbb{K}[\epsilon]/(\epsilon^2)) is the base space for 1-dimensional first order deformations. Similarly, Spec(𝕂[ϵ]/(ϵ n+1))Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1})) is the base space for 1-dimensional nn-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)Spec(A) is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is AA. By this reason spectra of Artinian local algebras are occasionally called fat points in the literature.


  • A classical example is the ring of dual numbers 𝕂[ϵ]/(ϵ 2)\mathbb{K}[\epsilon]/(\epsilon^2) over a field 𝕂\mathbb{K}.

  • Every prime power local ring is a local Artinian ring.

See also

commutative ringreduced ringintegral domain
local ringreduced local ringlocal integral domain
Artinian ringsemisimple ringfield
Weil ringfieldfield


Local Artinian \infty-algebras are discussed in

Last revised on January 12, 2023 at 18:49:51. See the history of this page for a list of all contributions to it.