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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.

Thanks for any help and suggestion!

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