The above answer says there is a ring such that for any nonmaximal prime ideal there is a nonmaximal prime ideal containing it. In fact valuation ring with infinite products of integers group as valuation group is an example. A paper about spectral space of commutative rings written by Hochster tells us there is also ring with spectral space which order just inverse the spectral space of a given ring. So there also exists a ring such that for any nonminimal prime ideal there exists another nonminimal prime ideal which containing is contained in it. That is there is a ring which has no prime with height 1.
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The above answer says there is a ring such that for any nonmaximal prime ideal there is a nonmaximal prime ideal containing it. In fact valuation ring with infinite products of integers group as valuation group is an example. A paper about spectral space of commutative rings written by Hochster tells us there is also ring with spectral space which order just inverse the spectral space of a given ring. So there also exists a ring such that for any nonminimal prime ideal there exists another nonminimal prime ideal which containing it. That is there is a ring which has no prime with height 1. |
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