$R$ such that $R/J$ is artinian ($J$ is the Jacobson radical) and there exists an idempotent cannot be lifted modulo $J$?

Question

I know a few examples of rings and ideals such that there exists an idempotent cannot be lifted modulo the ideal (for instance, $\mathbb Z$ and $n\mathbb Z$).

My question is: is there a ring $R$ such that $R/J$ is artinian ($J$ denotes Jacobson radical of $R$) and there exists an idempotent $e$ that cannot be lifted modulo $J$?

I can't come up with this case. Maybe you know an example?

Answer 1

Yes, in other words, there are semilocal rings which aren't lift/rad.

One such example is the semilocalization of $\mathbb Z$ at the complement of $S=(2)\cup (3)$.

In this ring, $R/J(R)\cong F_1\times F_2$ for two fields. Obviously this is artinian, and the nontrivial idempotents can't be lifted to $\mathbb Z$.