semisimple module

Definition

An R-module is called semisimple if it is isomorphic to a direct sum of simple submodules.

Equivalent conditions

The following conditions on an -module are equivalent:

1. is a sum (not necessarily direct) of simple submodules
2. is semisimple
3. For every submodule , there exists a submodule such that

Importance

This equivalence says that the direct part of the definition is technically not necessary (but can be convenient).

It also implies that complements of submodules, i.e. splittings of all submodules is an important structure in determining semisimplicity.

Proof

todo - Lecture 22

Morphism between semisimple modules

For two semisimple -modules then

Therefore,

Thus, using Schur’s Lemma, we have

Applying this to endomorphisms of a module , we see that

where each is simple. Then is a division ring, and

(Note, you can define a similar thing for ).

As a nice subcategory

Any submodule or quotient of a semisimple -module is semisimple, thus

is a nice subcategory.

Proof

todo - Lecture 23

Examples

todo - Lecture 22 has good ones.