Editing Associative algebra (section)

=== Commutative case ===
As ''A'' is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose [[residue field]]s are algebras over the base field ''k''. Now, a [[reduced ring|reduced]] Artinian local ring is a field and thus the following are equivalent{{sfn|Waterhouse|1979|loc=§ 6.2|ps=none}}
# <math>A</math> is separable.
# <math>A \otimes \overline{k}</math> is reduced, where <math>\overline{k}</math> is some [[algebraic closure]] of ''k''.
# <math>A \otimes \overline{k} = \overline{k}^n</math> for some ''n''.
# <math>\dim_k A</math> is the number of <math>k</math>-algebra homomorphisms <math>A \to \overline{k}</math>.

Let <math>\Gamma = \operatorname{Gal}(k_s/k) = \varprojlim \operatorname{Gal}(k'/k)</math>, the [[profinite group]] of finite Galois extensions of ''k''. Then <math>A \mapsto X_A = \{ k\text{-algebra homomorphisms } A \to k_s \}</math> is an anti-equivalence of the category of finite-dimensional separable ''k''-algebras to the category of finite sets with continuous <math>\Gamma</math>-actions.{{sfn|Waterhouse|1979|loc=§ 6.3|ps=none}}