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

: We may prove this more generally. Let be simple submodules.

Claim: There is a subset such that

Let be the maximal subset for which the sum is direct (not necessarily that , this will come later). Consider for some such that ,

Note that this is a submodule of (which is simple), so it must either zero - but then it could be added to as part of the direct sum, contradicting maximality - or all of . Thus, and

: Let . Let be the maximal subset such that

is direct (again, not necessarily that it will be all of ). Using the same technique as above we have that

therefore, we may take .

: First, we will prove that every non-zero module in contains a simple submodule. Take a non-zero element . Consider which is a submodule. The module homomorphism

then the kernel is a left ideal . By the first isomorphism theorem we have

Chose a maximal ideal that contains (this exists by Zorn’s lemma). We may take the quotient , which is a maximal submodule. Therefore, is maximal.

By the assumption of (3), we have

Therefore, looking we have (since )

is maximal in so we have

which must be simple. If it weren’t simple, then it would contain a non-trivial submodule which would contradict the maximality of .

Finally, let

by assumption

If then is non-zero and thus contains a simple submodule by the above which is a contradiction of the definition of .

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

Submodules: Say is a semisimple module with a submodule. Consider the submodule (of )

This is a semisimple submodule by construction (and the equivalence above). Also by the equivalence of statements above,

Note that any element can be written uniquely as

That gives because and is a submodule and closed under addition.

We may now consider

If there is a non-zero element then it is in a simple submodule, but then it would be contained in which is a contradiction. Therefore, and so is semisimple.

Quotients: Let for some submodule . Then we have

By the above, both and are semisimple, and by the first isomorphism theorem, we have that

which is semisimple.