A quotient and full matrix rings of which both Boolean

Question

Let $R$ be a ring with unity such that both $R/Soc(R_R)$ and the full matrix ring $M_n(R/Soc(R_R))$ are both Boolean rings ($n> 1)$. Is it true that $R$ is semisimple?

For semisimple rings the answer is obviously in affirmative, since in this case we have $R=Soc(R_R)$, and both rings are zero.

Thanks for any suggestion/help!

Answer 1

If $M_n(R)$ is Boolean for even a single $n>1$, then $R=\{0\}$. Otherwise the matrix ring is not commutative and contains nonzero nilpotent elements, neither of which work in a Boolean ring.

So yes, $R$ would necessarily be semisimple, in a trivial way.