Given an $R$-module $M$, then $soc(M) = ann_M(J(R))$ if $R/J(R)$ is semisimple

Question

Given an $R$-module $M$:

1. $J(R)$ is the Jacobson radical of $R$.
2. Given $X \subset R$, $ann_M(X) = \{m \in M: \forall r \in X, rm = 0\}$.
3. $Soc(M)$ is the socle of $M$ (the submodule of $M$ generated by all simple submodules of $M$).

The exercise ask to prove that $Soc(M) \subseteq ann_M(J(R))$ and that we have equality if $R/J(R)$ is semisimple. I was able to prove $Soc(M) \subseteq ann_M(J(R))$ but I am completely clueless on how to prove the other direction. What I tried so far without success:

1. Proving that $ann_M(J(R))$ is a semisimple module

2. For any $m \in ann_M(J(R)$, $\langle m\rangle$ is a simple module.

Any tips on how to approach this problem would be greatly appreciated.