When every ideal containing $J(R)$ is an intersection of maximal ideals

Question

Let $R$ be a commutative ring with $1$ such that every ideal containing $J(R)$, the intersection of all maximal ideals, is an intersection of maximal ideals. Is there any characterization for such a ring or is there any geometric interpretation for this property?

Answer 1

The condition is equivalent to $R/J$ being von Neumann regular.

Set $S = R/J$. Then the condition requires that no factor of $S$ have any nilpotent elements, and thus the square of any ideal in $S$ is the ideal itself. In particular, $rS = (rS)^2 = r^2 S$ for all $r \in S$, and thus $r = r^2 x$ for some $x \in S$, proving $S$ is regular. The converse is trivial.