A ring $R$ is termed as uniquely clean if each $r\in R$ has a unique representation $r=u+e$, where $u$ is a unit and $e$ is an idempotent. A good resource available in this regard may be this article. I am searching for a uniquely clean ring $R$ with the following conditions:
$$Soc(R_R)=J(R)\neq Soc(_RR),$$ where $Soc(R_R)$ and $Soc(_RR)$ stand, respectively, for the right and left socles of $R$, and $J(R)$ is the Jacobson radical of $R$.
Thanks for any help!