I am looking for an equivalent condition for a commutative ring $R $ with 1 to have the following property:
The only prime ideals contain $J (R) $, jacobson radical of $R $, are maximal ideals.
You are asking simply that $R/J(R)$ has Krull dimension zero.
This occurs exactly when $R/J(R)$ is von Neumann regular.
It follows from a more general theorem that says this:
In a commutative ring $R$, the following are equivalent:
- $R$ is zero dimensional
- $J(R)$ is a nil ideal and $R/J(R)$ is von Neumann regular.
Of course in your case, $J(R/J(R))$ is nil (since it is zero.)