Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a singly generated 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 $\prod V_i$

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 $\prod V_i$