A ring is called clean if each element of which is a sum of a unit and an idempotent. Recently, I asked a [question](https://mathoverflow.net/questions/267367/a-boolean-quotient-ring-of-a-prime-ring) about a prime ring $R$ in which $R/Soc(R\_R)$ is Boolean where the right socle $Soc(R\_R)$ is non-evident. 


My question is whether the ring $R$ in the comment by David Handelman a clean ring?


It is well-known that the ring $E$ in the comment is clean. So, if any element of $R$ which is a unit in $E$ is also a unit in $R$, then $R$ would be clean too.


Thanks for any suggestion and/or answer!



A prime ring which is or is not clean

> 
> So, if any element of $R$ which is a unit in $E$ is also a unit in $R$, then $R$ would be clean too.
> 
> 
> 


Well, that is easy to prove, right? Let $1+x\in R$ be a unit in $E$ ($x\in S$ of course.) Then there exists $y\in E$ such that $(1+x)y=1$. Then $y=1-xy\in R$ since $xy\in S$.


