semisimple ring

Definition

A ring is called semisimple if every R-module is semisimple.

Equivalent conditions

Let be a ring. The following are equivalent:

1. as a left module is semisimple.
2. Every -module is semisimple
3. Every -module is projective
4. Every -module injective.

Proof

todo - Lecture 23

As direct summand of regular -mod

Let be a semi-simple ring. Then every simple -module appears as a direct summand of as a left -module (i.e. the left regular -module).

Proof

Let be a simple -module and be non-zero. This gives the submodule is equal to since is simple. Thus, since is semi-simple, then it is a sum of simple modules as a left -module

By Schur’s Lemma,

therefore, for some since they all can’t be 0 because the map from to is not the zero map.

Examples

Example