Editing Ring (mathematics) (section)

=== Semisimple rings ===
{{Main|Semisimple module}}

A ''[[semisimple module]]'' is a direct sum of simple modules.  A ''[[semisimple ring]]'' is a ring that is semisimple as a left module (or right module) over itself.

====Examples====
* A [[division ring]] is semisimple (and [[simple ring|simple]]).
* For any division ring {{mvar|D}} and positive integer {{mvar|n}}, the matrix ring {{math|M{{sub|''n''}}(''D'')}} is semisimple (and [[simple ring|simple]]).
* For a field {{mvar|k}} and finite group {{mvar|G}}, the group ring {{math|''kG''}} is semisimple if and only if the [[characteristic (algebra)|characteristic]] of {{mvar|k}} does not divide the [[order (algebra)|order]] of {{mvar|G}} ([[Maschke's theorem]]).
* [[Clifford algebra]]s are semisimple.

The [[Weyl algebra]] over a field is a [[simple ring]], but it is not semisimple.  The same holds for a [[differential operator#Ring of multivariate polynomial differential operators|ring of differential operators in many variables]].

====Properties====
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)

For a ring {{mvar|R}}, the following are equivalent:
* {{mvar|R}} is semisimple.
* {{mvar|R}} is [[artinian ring|artinian]] and [[semiprimitive ring|semiprimitive]].
* {{mvar|R}} is a finite [[direct product]] <math display="inline"> \prod_{i=1}^r \operatorname{M}_{n_i}(D_i) </math> where each {{math|''n''{{sub|''i''}}}} is a positive integer, and each {{math|''D''{{sub|''i''}}}} is a division ring ([[Artin–Wedderburn theorem]]).

Semisimplicity is closely related to separability. A unital associative algebra {{mvar|A}} over a field {{mvar|k}} is said to be [[separable algebra|separable]] if the base extension <math>A \otimes_k F</math> is semisimple for every [[field extension]] {{math|''F'' / ''k''}}. If {{mvar|A}} happens to be a field, then this is equivalent to the usual definition in field theory (cf. [[separable extension]].)