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, noetherian, etc...?
Examples for some obvious constraints -
- Fields always have just one prime ideal - 0.
- Local rings have just one maximal ideal.
- Noetherian rings have only a finite number of minimal primes, and no infinitely ascending or descending sequence of prime ideals.
- Every ring has a minimal prime and a maximal prime.
(Question originally posted in m.se https://math.stackexchange.com/questions/1628013/partial-orders-that-can-be-realized-by-prime-ideals-of-commutative-rings)