Wedderburn-Artin theorem

If $R$ is a left semisimple ring, then

R\cong\mathbb{M}_{n_{1}}(D_{1})\times\cdot\cdot\cdot\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 $D_{i}$ , $i=1,2,\ldots,r$ . 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	Wedderburn-Artin theorem
Canonical name	WedderburnArtinTheorem
Date of creation	2013-03-22 14:19:08
Last modified on	2013-03-22 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	Artin-Wedderburn theorem
Related topic	SemiprimitiveRing