An elementary exercise from modules over algebras I was trying to prove following equivalence of statements: let $V$ be an $A$-module, $A$ is an algebra of finite dimension over a field $F$. Then following are equivalent:
1) $V$ is completely reducible $A$-module (i.e. it is direct sum of simple $A$-submodules).
2) $J(A)V=0$, where $J(A)$ is Jacobson radical of $A$.
I didn't get any direction to proceed; any hint?

If $J(R)$ annihilates $V$, then $V$ is also an $R/J(R)$ module. 
Since $R/J(R)$ is Artinian, it is a semisimple ring, so $V$ is a semisimple $R/J(R)$ module. But $R$ and $R/J(R)$ share the same simple modules, so $V$ is also a semisimple $R$ module.
For the other direction, I think you've already seen that $J(R)$, which annihilates all simple $R$ modules, obviously annihilates all semisimple $R$ modules.
