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.