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This is the first half of the question introduced in https://math.stackexchange.com/questions/258893/representations-of-direct-sums-of-matrix-algebras Let $A_1, A_2....A_n$ be n algebras with units $u_1, u_2,...u_n$ respectively. Let $A = A_1 \bigoplus A_2 \bigoplus....\bigoplus A_n$. Show that a representation $V$ of $A$ is irreducible if and only if $u_iV$ is an irreducible representation of $A_i$ for exactly one $i \in \{1,2..n\}$, while $u_i V = 0$ for all other $i$.

It seems that this has something to do with the fact that an irreducible module associated with the representation cannot be written as the direct sum of two strictly smaller modules.

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I'll do the case $n=2$, the general case follows by induction. The key point to observe here is that $V\cong u_1V\oplus u_2V$ as an $A$-module, sending $v\in V$ to $(u_1v,u_2v)$. The inverse given by $(u_1v,u_2w)\mapsto u_1v+u_2w$. You should check that these are $A$-module homomorphisms. This implies that in case $u_1V\neq 0\neq u_2V$ there is a non-trivial proper subrepresentation (even a direct summand) $0\subsetneq u_1V\subsetneq V$. Thus, you only have $u_iV\neq 0$ for at most one $i$. Furthermore $A_i$-subrepresentations will also be $A$-subrepresentations (as all other $A_j$ act as $0$).

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