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Module decompositions and idempotents.

January 12, 2013

In this post, by a ring R it will be understood that the ring comes with a multiplicative identity 1. An R-module M 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 N of M, there exists a submodule N' of M such that M = N \oplus N'.

A ring R is said to be semi-simple if every R-module is semi-simple. For example, it is a fundamental theorem in the theory of group representation that for a finite group G and a field \mathbb{K} whose characteristic does not divide the order of G, the group ring \mathbb{K}G, which can be defined as the ring of functions \varphi : G \to \mathbb{K} 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 R, an element e \in R is called idempotent if e^2 = e. Two idempotents e,f \in R are orthogonal if ef = fe = 0. Here is where this comes from:

If M is an R-module such that M = \bigoplus_{j=1}^nM_j, then we have the canonical inclusion homomorphisms \iota_j : M_j \to M and projection homomorphisms \pi_j : M \to M_j. They satisfy

  1. \pi_j \circ \iota_j = 1_{M_j}
  2. \pi_k \circ \iota_j = 0 \in \text{Hom}(M_j, M_k) if k \neq j
  3. \iota_1 \circ \pi_1 + \cdots + \iota_n \circ \pi_n = 1_M.

Then in the ring \text{End}(M) := \text{Hom}(M,M) there are the associated pairwise orthogonal idempotents e_1, \ldots, e_n defined by e_j = \iota_j \circ \pi_j. Note that these idempotents satisfy \sum_{i=1}^ne_i = 1. Conversely, given a set of orthogonal idempotents e_1, \ldots, e_n \in \text{End}(M) satisfying \sum_{i=1}^n e_i = 1, define M_j' := e_j(M). These are easily seen to be submodules and for any x \in M, we have

x = 1x = \sum_{i=1}^ne_ix \in M_1' + M_2' + \cdots + M_n'.

Moreover, for some i,j, (1 \leq i,j \leq n), let y \in M_i' \cap M_j', i.e. y = e_i(a) = e_j(b) for some a,b \in M. Then

y = e_i(y) = e_i^2(a) = e_i(e_j(b)) = 0

so that M = \bigoplus_{i=1}^ne_i(M). Thus we obtain the

Theorem: There is a bijective correspondence between pairwise orthogonal idempotents \{e_1, \ldots, e_n\} \subset \text{End}(M) such that \sum_{i=1}^ne_i = 1 and direct sum decompositions of M. In particular, every submodule of M is of the form e(M) for some idempotent e \in \text{End}(M).

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

Let’s now consider R as a left module over itself. Then a (left) submodule of R is the same thing as a (left) ideal of R so this correspondence is also true for the decomposition of a ring as a direct sum of ideals. Moreover, for any \varphi \in \text{End}_RR, say \varphi(1) = r \in R, then we must have, for all x \in R, that \varphi(x) = \varphi(x.1) = x\varphi(1) = xr. Thus every endomorphism of R arises as right multiplication by an element of the ring and \text{End}_RR \simeq R.

What this tells us is

Corollary: If R is a semi-simple ring (so in particular it is a semi-simple module over itself), every ideal of R is of the form Re for some idempotent e \in R and there is a bijective correspondence between ideal decompositions R = \bigoplus_{j=1}^n I_j and sets of pairwise orthogonal idempotent elements in R 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 R- modules, the “direct sum” of rings is not a categorical coproduct. Indeed, if R = \bigoplus_{j=1}^n R_j where R_j are rings with identity, then the inclusions \iota_j : R_j \to R do not send the 1_{R_j}‘s to 1_R. However, if e \in R is a central idempotent, meaning that e commutes with every other element of R, then Re is a ring with identity e and we have the

Proposition: If R = R_1 \times R_2 \times \cdots R_n and e_j is the identity element of R_j, then \{e_1, \ldots, e_n\} is a set of pairwise orthogonal central idempotents summing to 1_R. Conversely, given a set \{e_1, \ldots, e_n\} of orthogonal central idempotents summing to 1_R, we get a decomposition R = R_1 \times R_2 \times \cdots \times R_n.

Let’s go back to R-modules now. The question of when these idempotents correspond to a decomposition into irreducible submodules motivates the next definition: An idempotent e \in \text{End}(M) is said to be primitive if e = e_1 + e_2 for two orthogonal idempotents e_1, e_2 implies that either e_1 = 0 or e_2 = 0. We have the

Proposition: A submodule e(M) of M for e an idempotent endomorphism is irreducible if and only if e is primitive.

Proof. If e(M) = M_1 \oplus M_2, then since it is an R-module, by the correspondence we know that there are orthogonal idempotents e_1, e_2 such that e_i(M) = M_i (i=1,2) and e_1 + e_2 = 1_M. But then e_1e + e_2e = e so it suffices to show that e_1e and e_2e are orthogonal idempotents. If v\in M is such that e(v) \in M_1, then since M_1 = e_1(M), there is an u \in M such that e(v) = e_1(u). Because e_1, e_2 are orthogonal, we get

  1. (e_2e)(v) = e_2e_1(u) = 0 so (e_2e)^2(v) = e_2e(v)
  2. (e_1e)^2(v) = e_1ee_1e_1(u) = e_1ee_1(u) = e_1ee(v) = e_1e(v)
  3. e_1ee_2e(v) = e_1ee_2e_1(u) = 0 and
  4. e_2ee_1e(v) = e_2ee_1e_1(u) = e_2ee(v) = e_2e(v) = e_2e_1(u) = 0.

Similarly, if e(v) \in M_2, we get the same relations thus proving that e_1e and e_2e are orthogonal idempotents. So if e(M) is irreducible, one of M_1 or M_2 is \{0\} so one of e_1 or e_2 must be identically zero.

Conversely, if e = e_1 + e_2 for e_1, e_2 two orthogonal idempotents. Then for x \in e(M), x = e(m) = e_1(m) + e_2(m) \in (e_1 + e_2)(M). Also for y \in e_1(M), say y = e_1(z), we have e(e_1(z)) = e_1(e_1(z)) + e_2(e_1(z)) = e_1(z) = y. Similarly e_2(M) \subset e(M) and we have shown that e(M) = e_1(M) + e_2(M). Now if x \in e_1(M) \cap e_2(M), then for some a,b \in M we get x = e_1(a) = e_2(b) so x = e_1^2(a) = e_1(e_2(b)) = 0 and e(M) = M_1 \oplus M_2.

We thus get the important

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

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