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

(): We can make a free resolution of which gives that is a quotient of a [[202409302304|free -module]]. Thus we have

Since is a free module (by assumption of 1) then because semisimple modules form a nice subcategory then we have that is semisimple.

(): Duh…

(): Let there be modules , , and . Given the diagram

where is surjective. The image is a submodule, so by assumption of 2 there is a direct sum

Therefore, there is a splitting such that . Therefore, we may take . Thus, we have , so is projective.

(): This is basically the same thing: Now we have the diagram

Since the by assumption, we have .

(): Let be an -module with submodule . Since (by assumption) is injective, then we have the diagram

So there is a splitting and so is semisimple.

(): Same idea as the one above.

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.

Example

Let for some field . Then is semisimple as a left -module. We can see this since the “column space decomposition”

where is the projection onto the th factor (or column of the matrix). Note that each

which is simple.