As the title suggests, I'm curious if there is an example of a Baer ring $R$ such that the quotient ring of $R$ by its Jacobson radical, i.e. $R/J(R)$, is not Baer.
By "Baer ring" I mean a ring $R$ with identity such that the right annihilator of any nonempty subset $X\subseteq R$ is generated by an idempotent of $R$. That is, for any nonempty $X\subseteq R$ there exists $e=e^2\in R$ such that $\{a\in R\mid Xa=0\}=eR$.