# Partial Orders realized by Prime Ideals on 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, noetherian, 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.

(Question originally posted in m.se https://math.stackexchange.com/questions/1628013/partial-orders-that-can-be-realized-by-prime-ideals-of-commutative-rings)

Theorem (Hochster, Proposition 12 of this paper): Let $X$ be a poset. Then $X$ is isomorphic to the poset of prime ideals of a commutative ring iff it is order-isomorphic to a closed subset of $\{0,1\}^V$ for some set $V$ (putting the discrete topology on $\{0,1\}$ and the product topology on $\{0,1\}^V$, and the usual order on $\{0,1\}$ and the product order on $\{0,1\}^V$).
(Given an order-isomorphism $X\cong \operatorname{Spec}(R)$, the set $V$ can be taken to be the set of compact open subsets in the Zariski topology on $X$, and the Zariski topology can be recovered as the product topology on $\{0,1\}^V$, where you topologize $\{0,1\}$ such that $\{0\}$ is open and $\{1\}$ is not. Note that every poset embeds in some $\{0,1\}^V$, so the key point of this result is the requirement that it be a closed subset of $\{0,1\}^V$, which is a sort of compactness condition.)
• The closed ordered subspaces of $\{0,1\}^{V}$ are precisely the Priestley spaces. Priestley duality states that the Priestley spaces are precisely the spaces of prime ideals on bounded distributive lattices. Furthermore, De Groot duality gives a Stone-type duality between the category of all Priestley spaces and the category of all Zariski topologies on commutative rings. Jan 29, 2016 at 16:24