We take the ring $R:=\mathrm{M}_2(\Bbb C)$, we regard it as a $R$-module over itself, and we take its simple $R$-module $K:=\begin{pmatrix} 0 & \Bbb C \\ 0 & \Bbb C\end{pmatrix} \leq \mathrm{M}_2(\Bbb C)$. If $0\neq \alpha \in R$ is not invertible matrix, we want to show that there exists an invertible matrix $\beta \in R$, such that $\alpha \beta \in K$.
To prove that, it might be useful the facts one can see: $\alpha \in \mathrm{GL_2}(\Bbb C) \iff R\alpha=R$ and $\forall y\in K\backslash\{0\}$ it is $Ry=K$.
But how could we go further?