If $A$ is a direct sum of matrix algebra over $C$, what are all finite dimensional simple $A$-modules?

Solution 1:

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.