Editing Semisimple module (section)

{{Short description|Direct sum of irreducible modules}}{{see also|Semisimple algebra}}
In [[mathematics]], especially in the area of [[abstract algebra]] known as [[module theory]], a '''semisimple module''' or '''completely reducible module''' is a type of module that can be understood easily from its parts.  A [[ring (mathematics)|ring]] that is a semisimple module over itself is known as an Artinian '''semisimple ring'''.  Some important rings, such as [[group ring]]s of [[finite group]]s over [[field (mathematics)|fields]] of [[characteristic zero]], are semisimple rings.  An [[Artinian ring]] is initially understood via its largest semisimple quotient.  The structure of Artinian semisimple rings is well understood by the [[Artin–Wedderburn theorem]], which exhibits these rings as finite [[direct product]]s of [[matrix ring]]s.

For a group-theory analog of the same notion, see ''[[Semisimple representation]]''.