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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.
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  • $\begingroup$ It looks like you posted the same question twice in the span of 10 minutes. Please edit the first question instead. Thanks $\endgroup$
    – rschwieb
    Jan 26, 2016 at 17:52
  • $\begingroup$ Oops, my app said it failed to upload. I deleted the other copy. $\endgroup$
    – Alon Navon
    Jan 26, 2016 at 17:55
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    $\begingroup$ 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. $\endgroup$
    – rschwieb
    Jan 26, 2016 at 21:42
  • $\begingroup$ 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. $\endgroup$
    – rschwieb
    Jan 26, 2016 at 22:16
  • $\begingroup$ I thought so originally, but the existence of clear criteria for topological spaces kind of raises my hope. $\endgroup$
    – Alon Navon
    Jan 26, 2016 at 22:18

1 Answer 1

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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.

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