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Let $R$ be an integral commutative ring with unit. If $R$ is noetherian, then every ideal has finite height, in particular, there exist minimal non-zero prime ideals if (and only if) $R$ is not a field. If $R$ is a Krull ring, then the minimal non-zero prime ideals form a basis of $Div(R)$, so again, minimal non-zero primes exist if and only if $R$ is not a field.

My question is the following: Can anyone give an example of an integral commutative ring which is not a field that has no minimal non-zero prime ideals, i.e. a ring in which every non-zero prime ideal properly contains another non-zero prime ideal?

Answer (provided by Anderson, Kang, Park: Anti-archimedean rings and power series rings, as Francesco Polizzi points out here, thanks to Georges for the reference): An anti-Archimedean valuation domain has no minimal non-zero (= height one) prime ideals.

concerning former edits:

(I recently edited to clarify that the question is meant to refer to integral domains. The problem with a proof using Zorns Lemma on the set of non-zero prime ideals would be that the intersection of a chain of non-zero primes might be zero.)

(I again edited to clarify that the question is meant to refer to integral domains which are not fields.)

I'm sorry for not being precise from the start, I will try to be more explicit when I post my next question. Both contribution from you were perfectly to the purpose, until I a added the further restrictions. The motivation for the question is the following: An integral domain which is also either noetherian or a Krull ring has minimal non-zero prime ideals if and only if it is not a field. However, this seems to be hardly enough evidence to suppose that such a statement should be true for all integral domains. So I wondered whether there exist counter examples. And indeed I learned from the first response that non-integral domains do provide counter examples.

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I have deleted my answer after your edit, as I see that it did not address what you had intended to ask. –  Manny Reyes Nov 6 '13 at 16:27
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If I find another counterexample, will you edit the question to specifically exclude it? :-) –  Bruno Joyal Nov 6 '13 at 16:46
    
No, if I should realize that I did not manage to formulate my question in a precise manner after three attempts, then I would give it up entirely. –  user105881 Nov 6 '13 at 19:36
    
Paul VanKoughnett claims here that there indeed exists a domain in which every non-zero prime has infinite height. Unfortunately no argument nor reference supports his assertion. –  Georges Elencwajg Nov 6 '13 at 20:17
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Update: Francesco Polizzi, an extremely gifted and reliable member of MathOverflow, gives a reference here confirming that there indeed exist domains in which every non-zero prime has infinite height. So that answers your question. –  Georges Elencwajg Nov 6 '13 at 20:59
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Answer (provided by Anderson, Kang, Park: Anti-archimedean rings and power series rings, as Francesco Polizzi points out here, thanks to Georges Elencwajg for the reference): An anti-Archimedean valuation domain has no minimal non-zero (= height one) prime ideals.

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