# Semisimple module

A **semisimple module** is, informally, a module that is not
far removed from simple modules.
Specifically, it is a module with the following property:
for every submodule , there exists a submodule
such that and , where by 0 we mean the
zero module.

## Classification of semisimple modules

It happens that semisimple modules have a convenient classification (assuming the axiom of choice). To prove this classification, we first state some intermediate results.

**Proposition.** Let be a semisimple left -module.
Then every submodule and quotient module of is also
simple.

*Proof.* First, suppose that is a submodule of .
Let be a submodule of , and let be a submodule of
such that and . We note that if
and are elements such that ,
then . It follows that
Since , it follows that is semisimple.

Now let us consider a quotient module of , with the canonical homomorphism . Let be a submodule of . Then is a submodule of , so there exists a submodule such that and . Then in , since is surjective, Therefore is semisimple as well.

**Lemma 1.** Let be a ring, and let be a nonzero
cyclic left -module. Then contains a
maximal proper submodule.

*Proof.* Let be a generator of . Let
be the set of submodules that avoid , ordered by inclusion.
Then is nonempty, as .
Also, if is a nonempty chain in ,
then is an element of ,
as this is a submodule of that does not contain .
Then is an upper bound on the chain
; thus every chain has an upper bound. Then by
Zorn's Lemma, has a maximal element.

**Lemma 2.** Every cyclic semisimple module has a simple submodule.

*Proof.* Let be a cyclic semisimple module, and let
be a generator for . Let be a maximal proper submodule of
(as given in Lemma 1), and let be a submodule such that
and . We claim that is simple.

Indeed, suppose that is a nonzero submodule of . Since the sum is direct, it follows that the sum is direct. Since strictly contains , it follows that , so ; it follows that ; thus is simple.

**Theorem.** Let be a left -module, for a ring . The
following are equivalent:

- is a semisimple -module;
- is isomorphic to a direct sum of simple left -modules;
- is isomorphic to an (internal) sum of -modules.