Let $R$ be a commutative ring with unity. I want a proof of the fact that
$R$ is zero-dimensional (in the sense that all prime ideals are maximal) if and only if $R/J(R)$ is von Neumann regular and $J(R)$ is nil, where $J(R)$ is the Jacobson radical of $R$.
If $J(R)$ is nil then it coincides with the nil radical $\sqrt 0$ of $R$. So, if $R/J(R)$ is regular we have $R/\sqrt 0$ regular which implies that $R$ is zero-dimensional by an easy argument. Now, the reverse implication is at stake. Thanks for any suggestion,