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.

Answer 2

The book "Algebra" by "Hungerford", page 437, Corollary 3.5: If $I$ is an ideal in a semisimple left Artinian ring $R$, then $I = Re$, where $e$ is an idempotent which is in the center of $R$.

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Edition by OP comment.
Assuming every maximal ideal is finitely generated, there is a positive answer to the question; see here