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(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.

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goblin GONE
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    So I assume you are here really interested in the case where $R$ is a domain (as otherwise we are just back in that other question). – Tobias Kildetoft Oct 29 '15 at 09:03
  • @TobiasKildetoft, true; if $R$ fails to be a domain then we already have an answer to this question. But I'll leave it to the answer-writers whether or not to include this assumption, and if so, where to include it, and for which results. Hopefully, this encourages people to give cleaner, more self-contained answers. But you're right, of course. – goblin GONE Oct 29 '15 at 09:06
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    Ohh, I should probably also mention that all Dedekind domains in fact have this property, extending the PID example, but I have no idea how "common" other examples might be. – Tobias Kildetoft Oct 29 '15 at 09:08
  • There is probably a good supply of interesting domains with Krull dimension 1 even beyond Dedekind domains, but I guess a characterization is not really possible... – rschwieb Oct 29 '15 at 10:58

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