(All my rings are commutative and unital.)

Here the following question is asked:

Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about the ring $R$?

Awesome, awesome question. However, the requirement that $\{0\}$ is maximal if it is prime is very strong. In particular, since $\{0_R\}$ is a prime ideal whenever $R$ is an integral domain, hence every integral domain satisfying the above condition will automatically be a field; this means, in particular, that $\mathbb{Z}$ does not satisfy the condition under question (which is a real shame). So, lets weaken it.

Question.Can we usefully characterize those rings whose every non-zero prime ideal is maximal?

I know that every PID has this property, and that this characterizes the PID's among the UFD's.