# Module decompositions and idempotents.

In this post, by a ring it will be understood that the ring comes with a multiplicative identity . An module is said to be *simple* if it has no non-zero proper submodule and *semi-simple* if it is a direct sum of simple submodules. This definition can be shown to be equivalent to the property that for every submodule of , there exists a submodule of such that .

A ring is said to be *semi-simple* if every module is semi-simple. For example, it is a fundamental theorem in the theory of group representation that for a finite group and a field whose characteristic does not divide the order of , the group ring , which can be defined as the ring of functions with convolution, is semi-simple. I’ll try to write up this story soon.

What I want to discuss today is a description of the irreducible components of a semi-simple ring in terms of certain elements of this ring called idempotents. This is again of fundamental importance in the theory of group representation and I want to understand this in order to understand the representation theory of the symmetric group.

Given a ring , an element is called *idempotent* if . Two idempotents are *orthogonal* if . Here is where this comes from:

If is an module such that , then we have the canonical inclusion homomorphisms and projection homomorphisms . They satisfy

- if
- .

Then in the ring there are the associated pairwise orthogonal idempotents defined by . Note that these idempotents satisfy . Conversely, given a set of orthogonal idempotents satisfying , define . These are easily seen to be submodules and for any , we have

Moreover, for some , , let , i.e. for some . Then

so that . Thus we obtain the

**Theorem:** There is a bijective correspondence between pairwise orthogonal idempotents such that and direct sum decompositions of . In particular, every submodule of is of the form for some idempotent .

Note that the last sentence is a direct consequence of the remark in the first paragraph of the post.

Let’s now consider as a left module over itself. Then a (left) submodule of is the same thing as a (left) ideal of so this correspondence is also true for the decomposition of a ring as a direct sum of ideals. Moreover, for any , say , then we must have, for all , that . Thus every endomorphism of arises as right multiplication by an element of the ring and .

What this tells us is

**Corollary:** If is a semi-simple ring (so in particular it is a semi-simple module over itself), every ideal of is of the form for some idempotent and there is a bijective correspondence between ideal decompositions and sets of pairwise orthogonal idempotent elements in such that their sum is equal to 1.

It is interesting to note that even though the direct sum is a biproduct in the category of modules, the “direct sum” of rings is not a categorical coproduct. Indeed, if where are rings with identity, then the inclusions do not send the ‘s to . However, if is a central idempotent, meaning that commutes with every other element of , then is a ring with identity and we have the

**Proposition:** If and is the identity element of , then is a set of pairwise orthogonal central idempotents summing to . Conversely, given a set of orthogonal central idempotents summing to , we get a decomposition .

Let’s go back to modules now. The question of when these idempotents correspond to a decomposition into irreducible submodules motivates the next definition: An idempotent is said to be *primitive* if for two orthogonal idempotents implies that either or . We have the

**Proposition:** A submodule of for an idempotent endomorphism is irreducible if and only if is primitive.

*Proof. *If , then since it is an module, by the correspondence we know that there are orthogonal idempotents such that and . But then so it suffices to show that and are orthogonal idempotents. If is such that , then since , there is an such that . Because are orthogonal, we get

- so
- and
- .

Similarly, if , we get the same relations thus proving that and are orthogonal idempotents. So if is irreducible, one of or is so one of or must be identically zero.

Conversely, if for two orthogonal idempotents. Then for , . Also for , say , we have . Similarly and we have shown that . Now if , then for some we get so and .

We thus get the important

**Corollary:** For an module, there is a bijective correspondence between decompositions of into irreducible submodules and sets of pairwise orthogonal primitive idempotents in the endomorphism ring of .

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