Schur’s lemma states that if is a simple module, then is a division ring. A similar easy argument shows that:
Example 6. For simple -modules we have
Let’s generalize Schur’s lemma: let be a finite direct product of simple -submodules. So where each is a simple -module and for all Therefore, by Example 6 and Theorem 1, where is a division ring by Schur’s lemma. An important special case is when is a semisimple ring. (Note that simple submodules of a ring are exactly minimal left ideals of that ring.)
Theorem 2. (Artin-Wedderburn) Let be a semisimple ring. There exist a positive integer and division rings such that .
Proof. Obvious, by Example 1 and the above discussion.
Some applications of Theorem 2.
1. A commutative semisimple ring is a finite direct product of fields.
2. A reduced semisimple ring is a finite direct product of division rings.
3. A finite reduced ring is a finite direct product of finite fields.