# Direct Sums of Matrix Algebra

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.

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 $0$).