Let $R$ be a ring without identity and $n$ be a positive integer. Prove that the ring $R$ is simple iff $M_n(R)$ is simple.
When $R$ is unital ring (i.e. with identity) it is easy because we know the structure of two-sided ideals in $M_n(R)$. So we can consider the case, when $R$ is non-unital.