If $A$ is a direct sum of matrix algebra, what are all finite-dim simple $A$-modules?
For simple case of $2$ matrix algebras, consider $A = M_n(\mathbb{C}) \oplus M_m(\mathbb{C})$. Suppose $M$ is a finite dimensional $A$-module, then it is automatically a finite dimensional $M_n(\mathbb{C}) $-module and a finite dimensional $M_m(\mathbb{C}) $-module. In particular, $M$ must be both a direct sum of $\mathbb{C}^n$ and of $\mathbb{C}^m$ (This is because, if $M$ is a $M_n(\mathbb{C})$-module, fix $ a_0 \in M$, then $\mathbb{C}^n \cong M_n(\mathbb{C})a_0$ is a simple $M_n(\mathbb{C})$-module, and we can choose $a_1 \notin M_n(\mathbb{C})a_0$ and decompose $M$ into a direct sum of $\mathbb{C}^n$).
Therefore, $$M = \bigoplus_j^N(\bigoplus_{1 \leq i \leq n} \mathbb{C}^m) = \bigoplus_j^N(\bigoplus_{1 \leq i \leq m} \mathbb{C}^n) = \bigoplus_j^N( \mathbb{C}^{m} \otimes_\mathbb{C} \mathbb{C}^{n}) = \bigoplus_j^N( \mathbb{C}^{mn}) = \mathbb{C}^{mnN} .$$
Let $M_m(\mathbb{C}) \oplus M_n(\mathbb{C})$ acts on $\mathbb{C}^{m} \otimes_\mathbb{C} \mathbb{C}^{n}$ naturally on each component, then it is easy to see the action is transitive and that it is stable under action, making it a simple $A$-module. Thus all $M$ is a direct sum of simple $A$-module $\mathbb{C}^{m} \otimes_\mathbb{C} \mathbb{C}^{n}$ and this classifies all simple $A$-modules. For any direct sum $A = \bigoplus_j M_{n_j}(\mathbb{C})$, the simple
$A$-module is therefore $M^{\Pi_jm_j}$. Is this reasoning correct?
Best Answer
Let me generalize the question. Suppose $A$ is the direct sum $R\oplus S$ of two rings (though I prefer the terminology/notation "direct product $R\times S$"), and let $M$ be a (right) $A$-module.
Let $e=(1_R,0)\in A$ and $f=(0,1_S)\in A$. Then $Me$ and $Mf$ are both $A$-modules, and $M = Me\oplus Mf$. $Me$ has a natural $R$-module structure, and $S$ acts on it as zero. $Mf$ has a natural $S$-module structure, and $R$ acts on it as zero.
So any $A$-module $M$ is the direct sum of an $R$-module and an $S$-module. In particular, the simple $A$-modules are just the simple $R$-modules (with $S$ acting as zero) and the simple $S$-modules (with $R$ acting as zero).
In the case $A=M_n(\mathbb{C})\oplus M_m(\mathbb{C})$, $A$ has two simple modules (up to isomorphism): the $n$-dimensional simple $M_n(\mathbb{C})$-module, with $M_m(\mathbb{C})$ acting as zero, and the $m$-dimensional simple $M_m(\mathbb{C})$-module, with $M_n(\mathbb{C})$ acting as zero.