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

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.
• It looks like you posted the same question twice in the span of 10 minutes. Please edit the first question instead. Thanks Jan 26, 2016 at 17:52
• Oops, my app said it failed to upload. I deleted the other copy. Jan 26, 2016 at 17:55
• You can really say something a lot stronger than $4$. Every prime ideal contains a minimal prime ideal, and is contained in a maximal prime ideal. Then the poset has a unique greatest element iff the ring is local. Domains would have unique minimal elements, but then so would any ring with a unique minimal prime ideal. Jan 26, 2016 at 21:42
• It's an interesting question but is probably too ambitious. If there were a method of building a ring with a particular poset of prime ideals, then we would be on better ground. Jan 26, 2016 at 22:16
• I thought so originally, but the existence of clear criteria for topological spaces kind of raises my hope. Jan 26, 2016 at 22:18