Endormorphism rings over simple modules 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!
 A: 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? 
