WedderburnArtin theorem
If $R$ is a left semisimple ring^{}, then
$$R\cong {\mathbb{M}}_{{n}_{1}}({D}_{1})\times \mathrm{\cdots}\times {\mathbb{M}}_{{n}_{r}}({D}_{r})$$ 
where each ${D}_{i}$ is a division ring and ${\mathbb{M}}_{{n}_{i}}({D}_{i})$ is the matrix ring over ${D}_{i}$, $i=1,2,\mathrm{\dots},r$. The positive integer $r$ is unique, and so are the division rings (up to permutation^{}).
Some immediate consequences of this theorem:

•
A simple (http://planetmath.org/SimpleRing) Artinian ring is isomorphic^{} to a matrix ring over a division ring.

•
A commutative^{} semisimple ring is a finite direct product^{} of fields.
This theorem is a special case of the more general theorem on semiprimitive rings.
Title  WedderburnArtin theorem 

Canonical name  WedderburnArtinTheorem 
Date of creation  20130322 14:19:08 
Last modified on  20130322 14:19:08 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  14 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 16D70 
Synonym  structure theorem on semisimple rings 
Synonym  ArtinWedderburn theorem 
Related topic  SemiprimitiveRing 