Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a  complex algebra. 

Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible submodules $V_i\leq V$ satisfying the following conditions? 

1) $V_i\cap V_j=0$, 
2) $V$ is embedded into $\prod V_i$