Nilpotent or just nil idempotent ideal?

Question

Let $R$ be a ring with zero Krull dimension and $I$ be an idempotent ideal contained in the Jacobson radical $J(R)$ of $R$. Could one infer just with these hypotheses that $I$ is a nilpotent ideal?

I know that Krull dimension being zero implies that $J(R)$ is equal to the set of nilpotent elements of $R$ (whence a nil ideal), and also $R/J(R)$ is von-Neumann regular. So, I only deduce that $I$ is a nil ideal.

Thanks for any hint or suggestion!

Answer 1

This example works also here. The only thing to show is $\dim R=0$, but this is obvious.