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The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.

My question was inspired from Exercise 1.8 of Atiyah-Macdonald's book in Commutative Algebra:

The set of prime ideals of a nonzero ring has minimal elements with respect to inclusion.

Thus the above is also a necessary condition. My question is if there are more necessary conditions which are also sufficient, i.e.,

Are there necessary and sufficient conditions that a poset must possess so that it is the poset of a ring's spectrum.

Answers (possibly partial answers) are welcome for both necessary and sufficient conditions as well as imposing restrictions on the ring. For example, the poset corresponding to Noetherian rings must obey the ACC. If the poset is a lattice, then the ring must be local. Do nice conditions occur assuming our ring is Artinian or a Dedekind domain?

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József Pelikán told me that every finite poset can be realized as the spectrum of a ring (although I don't have a reference). A crucial point is that you can obtain closed and open subspaces by ring quotients and localizations, respectively. So the real challenge is building a ring with a big, complicated spectrum with the posets you want as subspaces. I enjoyed constructing examples by hand for some small posets; it's a nice exercise. –  Gene S. Kopp Apr 29 '11 at 7:51
    
@Kopp, The papers below due to Hochster and Lewis do have results for finite posets. Although it will take time for me to read the papers, I had a glance. –  Abhishek Parab Apr 29 '11 at 10:22
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2 Answers 2

up vote 10 down vote accepted

Hochster answered this question in his thesis (by finding such conditions). See: Hochster, M. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 1969 43–60.

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Actually, Hochster gave a positive answer only for finite posets, because a finite poset is the specialisation relation of a (finite) spectral space. The full question on the type of posets that are realized by spectra of rings is open and called the "Kaplansky problem". It was asked by Kaplansky in the book Commutative rings, Allyn and Bacon Inc., p. x+180 (1970) (see page 5) –  Marcus Nov 6 '12 at 23:17
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Beside considering the article by Hochster you could also take a look at

WILLIAM J. LEWIS, The Spectrum of a Ring as a Partially Ordered Set, JOURNAL OF ALGEBRA 25, 419-434 (1973).

It is online:

www.maths.manchester.ac.uk/.../Lewis-Thespectrumofaringasapartiallyorderedset.pdf

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