If $R$ is a unital ring, it is well-known that its Jacobson radical $J(R)$ contains no non-zero idempotent element of $R$. My question:
Is there a ring $R$ such that $J(R)$ contains a non-zero idempotent ideal of $R$?
If $R$ is Noetherian, then each ideal is finitely generated. So, in case $I\subseteq J(R)$ with $I=I^2$ we would have $I=I^2\subseteq IJ(R)\subseteq I$. Now, by Nakayama's lemma we get $I=0$.
But, I do not have any idea in general case. Thanks for any help!