Editing Associative algebra (section)

== Finite-dimensional algebra ==
{{See also|Central simple algebra}}

Let ''A'' be a finite-dimensional algebra over a field ''k''. Then ''A'' is an [[Artinian ring]].

=== 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}}

=== Noncommutative case ===
Since a [[simple Artinian ring]] is a (full) matrix ring over a [[division ring]], if ''A'' is a simple algebra, then ''A'' is a (full) matrix algebra over a division algebra ''D'' over ''k''; i.e., {{nowrap|1=''A'' = M<sub>''n''</sub>(''D'')}}. More generally, if ''A'' is a semisimple algebra, then it is a finite product of matrix algebras (over various division ''k''-algebras), the fact known as the [[Artin–Wedderburn theorem]].

The fact that ''A'' is Artinian simplifies the notion of a [[Jacobson radical]]; for an Artinian ring, the Jacobson radical of ''A'' is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)

The '''Wedderburn principal theorem''' states:{{sfn|Cohn|2003|loc=Theorem 4.7.5|ps=none}} for a finite-dimensional algebra ''A'' with a [[nilpotent ideal]] ''I'', if the projective dimension of {{nowrap|''A'' / ''I''}} as a module over the [[enveloping algebra of an associative algebra|enveloping algebra]] {{nowrap|(''A'' / ''I'')<sup>e</sup>}} is at most one, then the natural surjection {{nowrap|''p'' : ''A'' → ''A'' / ''I''}} splits; i.e., ''A'' contains a subalgebra ''B'' such that {{nowrap|''p''{{!}}<sub>''B''</sub> : ''B'' {{overset|lh=0.5|~|→}} ''A'' / ''I''}} is an isomorphism. Taking ''I'' to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of [[Levi's theorem]] for [[Lie algebra]]s.<!--
A finite-dimensional algebra ''A'' is called a [[split algebra]] if each endomorphism of a simple ''A''-module is given by a scalar multiplication. Equivalently,

For example, a finite-dimensional algebra is a split when the base field is algebraically closed.-->