I have been struggling with this Representation Theory question for the past week:

Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomorphic to $S_j$ whenever $i \neq j$ and fix positive integers $n_1, \ldots, n_r$. Show that we have ring isomorphisms

\begin{align*} End_R{({S_1}^{n_1} \oplus \ldots \oplus {S_r}^{n_r} )} & \cong End_R{({S_1}^{n_1})} \times \ldots \times End_R{({S_r}^{n_r})} \\ & \cong {M_{n_1}}{(End_R{(S_1)})} \times \ldots \times {M_{n_r}}{(End_R{(S_r)})} \end{align*}

If anyone can help, I would highly appreciate it. I think that to prove the second isomorphism, it might be useful to first show that $ End_R{({S_i}^{n_i})} \cong {M_n}{(End_R{(S_i)})}$, which I am not sure how to show or if it even helps. For the first part, there is a lemma we have proved in lectures that looks useful: If $M$ is an $R$-module and $V_1 , V_2$ are simple submodules with $V_1 \ncong V_2$, then $$ End_R{(V_1 \oplus V_2)} \cong End_R{(V_1)} \times End_R{(V_2)} .$$ I am not sure how to use this though, as ${S_i}^{n_i}$'s are not simple. Anyone could help me show both isomorphisms of my initial question? We have also covered Artin Wedderburn Decomposition and 3 versions of Schur's Lemma in lectures. Thanks for your help!


The first step is to adapt the proof that $$End_R(V_1\oplus V_2)\cong End_R(V_1)\oplus End_R(V_2)$$ to the case of arbitrary $V_1$ and $V_2$ satisfying $Hom_R(V_1,V_2)=0$.

Then, prove the following:

For $\phi\in\mathrm{End}_R(S^n)$, define $$\phi_{ij}=\pi_j\circ\phi\circ \iota_i$$ where $\pi_j:S^n\to S$ is the projection onto the $j$th factor and $\iota_i:S\to S^n$ is the natural embedding into the $i$th factor. Then, the map $$\mathrm{End}_R(S^n)\to M_n(\mathrm{End}_R(S))$$ given by $\phi\mapsto(\phi_{ij})_{i,j=1}^n$ is an isomorphism.


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