Let $R$ be a Noetherian semi-local (https://en.m.wikipedia.org/wiki/Semi-local_ring) ring with Jacobson radical $J(R)$. If $M$ is a finitely generated left-$R$-module such that $ann_R(M) \subseteq J(R)$, then for every simple left $R$-module $S$, does there exist a surjection $M\to S\to 0$ ?
Thoughts on the Commutative case: When $R$ is commutative, this holds true: The inclusion $ann(M) \subseteq J(R)$ implies every maximal ideal $\mathfrak m$ contains $ann(M)$ so every maximal ideal is in $Supp(M)$, so $M_{\mathfrak m}\neq 0$ , so by Nakayama Lemma $M_{\mathfrak m}\neq (\mathfrak m R_{\mathfrak m}) M_{\mathfrak m}$, consequently $M \neq \mathfrak m M$. As $R$ is commutative, so every $R/\mathfrak m$ is a field, so $M /\mathfrak m M$ is a non-zero $R/\mathfrak m$-vectorspace. Now the surjection $M \to M/\mathfrak m M\to 0$ gives a surjection $M \to R/\mathfrak m$.
Let $R$ be the ring $\begin{pmatrix} k & k\\0 & k \end{pmatrix}$ of upper triangular $2\times 2$ matrices over a field $k$, and let $M$ be the left module $\begin{pmatrix}k\\k\end{pmatrix}$ of column vectors.
Then the annihilator of $M$ is zero, but $M$ does not have the simple module $\begin{pmatrix}k\\0\end{pmatrix}$ as a quotient.