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I know that if $U$ is a simple left $R-$module then $\text{End}_R(U)$ is a division ring. It is an easy argument. But I'm struggling to see what happens when we move up to trying to classify endomorphism rings of semisimple modules. Even worse, I don't quite understand how to approach $\text{End}_R(U\oplus V)$ where $U$ and $V$ are both simple left $R-$modules. Any ideas on how to approach this would be much appreciated. Thanks!

J. Moeller
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  • The Artin-Wedderburn theorem tells you essentially that the endomorphism rings of finite length semisimple modules are exactly the semisimple rings. Endomorphism rings of semisimple modules in general are always von Neumann regular rings. But not every VNR ring is the endomorphism ring of a semisimple module. So classifying is not easy (it isn’t even easy for division rings) but you have the VNR result at least! – rschwieb Nov 27 '17 at 11:35

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In general, $\text{Hom}(-, -)$ respects direct sums in both variables, meaning that we have

$$\text{Hom}(U \oplus V, W) \cong \text{Hom}(U, W) \oplus \text{Hom}(V, W)$$

and

$$\text{Hom}(W, U \oplus V) \cong \text{Hom}(W, U) \oplus \text{Hom}(W, V).$$

You should prove this if you haven't already. Together with some compatibilities between these isomorphisms and composition, it follows that if $U$ and $V$ are any two modules whatsoever, $\text{End}(U \oplus V)$ is the "matrix algebra"

$$\left[ \begin{array}{cc} \text{End}(U) & \text{Hom}(V, U) \\ \text{Hom}(U, V) & \text{End}(V) \end{array} \right]$$

(by which I mean it's the direct sum of all of these things as an abelian group, and the multiplication is given by "matrix multiplication").

If $U$ and $V$ are furthermore simple, then there are two cases depending on whether or not $U$ and $V$ are isomorphic. The easier case is when they are not isomorphic; in that case $\text{Hom}(V, U) = \text{Hom}(U, V) = 0$ and we get $\text{End}(U \oplus V) \cong \text{End}(U) \times \text{End}(V)$. Can you finish from here, including the other case?

Qiaochu Yuan
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  • This is the kind of answer I was looking for. I feel like I have some tools that I can use, now. Thankyou! – J. Moeller Nov 27 '17 at 00:36
  • One of the potions switches sums to products, right? That would be useful if the poster wants to go on to handle non finitely generated semisimple modules. As it is, they might think it extends to any number of summands .You can definitely remember which position it is faster than I... – rschwieb Nov 27 '17 at 02:06
  • The infinite case is slightly delicate; if the OP is interested they can ask about that separately. – Qiaochu Yuan Nov 27 '17 at 02:07