Artin-Wedderburn theorem

Statement

A semisimple ring is isomorphic to a product of matrix algebras of division rings.

Proof

Let be a semisimple ring. as a left -module then is semisimple, so by Schur’s Lemma we have that

where

is the semisimple decomposition of , that is are all simple modules, and is a division algebra.

By the equivalence from and the opposite ring

This gives

Corollaries

• Commutative semi-simple algebras are products of fields.