Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent?
I know only that if each maximal ideal is generated by an idempotent element then $R$ turns out to be semisimple Artinian. I think this fact is associated with my question, at least if one could show that any maximal ideal is generated by an idempotent element.
Thanks for any suggestion!