Editing Associative algebra (section)

=== 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.-->