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 $M$.