Editing Semisimple module (section)

=== Examples ===
* For a [[commutative ring]], the four following properties are equivalent: being a [[semisimple ring]]; being [[Artinian ring|artinian]] and [[reduced ring|reduced]];{{sfn|ps=|Bourbaki|2012|p=133|loc=VIII}} being a [[reduced ring|reduced]] [[Noetherian ring]] of [[Krull dimension]] 0; and being isomorphic to a finite direct product of fields.
* If ''K'' is a field and ''G'' is a finite group of order ''n'', then the [[group ring]] ''K''[''G''] is semisimple if and only if the [[characteristic (algebra)|characteristic]] of ''K'' does not divide ''n''. This is [[Maschke's theorem]], an important result in [[group representation theory]].
* By the [[Wedderburn–Artin theorem]], a unital ring ''R'' is semisimple if and only if it is (isomorphic to) {{nowrap|M<sub>''n''<sub>1</sub></sub>(''D''<sub>1</sub>) × M<sub>''n''<sub>2</sub></sub>(''D''<sub>2</sub>) × ... × M<sub>''n''<sub>''r''</sub></sub>(''D''<sub>''r''</sub>)}}, where each ''D''<sub>''i''</sub> is a [[division ring]] and each ''n''<sub>''i''</sub> is a positive integer, and M<sub>''n''</sub>(''D'') denotes the ring of ''n''-by-''n'' matrices with entries in ''D''.
* An example of a semisimple non-unital ring is M<sub>∞</sub>(''K''), the row-finite, column-finite, infinite matrices over a field ''K''.