[Math] Idempotent ideals in certain commutative rings

Question

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!

Answer 1

In a commutative ring $R$ every ideal is idempotent (iff every ideal is radical) iff $R$ is VNR.

Then the question asks if a commutative ring $R$ with $J(R)=0$ and $\mathfrak m^2=\mathfrak m$ for every maximal ideal $\mathfrak m$ is VNR.

The answer is negative: the ring of continuous functions $R=\mathcal C[0,1]$ satisfies both conditions and it's not VNR.