A non-clean ring $R$ with $R/Soc(R_R)$ Boolean

Question

I am searching for a non-commutative ring $R$ with identity such that $R$ is not a clean ring and $R/Soc(R_R)$ is a Boolean ring. By a clean ring I mean a ring each of whose elements is a sum of a unit and an idempotent, and by Boolean ring I mean a ring each of whose elements is an idempotent. I have run into this article Example 4, but, $R/Soc(R_R)$ is not Boolean, though $R$ is not clean.

Thanks for any help and suggestion!