# 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:

This theorem is a special case of the more general theorem on semiprimitive rings.

Title Wedderburn-Artin theorem WedderburnArtinTheorem 2013-03-22 14:19:08 2013-03-22 14:19:08 CWoo (3771) CWoo (3771) 14 CWoo (3771) Theorem msc 16D70 structure theorem on semisimple rings Artin-Wedderburn theorem SemiprimitiveRing