Partial orders that can be realized by prime ideals of commutative rings

Question

Is there a general criterion for which partial orders can be realized by the prime ideals of commutative rings (like we have for topological spaces - https://en.wikipedia.org/wiki/Spectral_space)?

And in general, what constraints on the partial order are created by additional classical properties of rings - such as being a UFD, principal domain, etc...?

Examples for some obvious constraints -

1. Fields always have just one prime ideal - 0.
2. Local rings have just one maximal ideal.
3. Noetherian rings have only a finite number of minimal primes, and no infinitely ascending or descending sequence of prime ideals.
4. Every ring has a minimal prime and a maximal prime.

Answer 1

Question answered by Eric Wofsey here:

https://mathoverflow.net/questions/229611/partial-orders-realized-by-prime-ideals-on-commutative-rings.

Apparently there is such a criterion. See: https://en.wikipedia.org/wiki/Priestley_space.