TITLE: Are there one-dimensional ideals in any local ring
QUESTION [6 upvotes]: I would like to know if it is always possible to find a one-dimensional ideal in a local commutative ring... actually I am interested in finding a curve through a point on a scheme (locally). If the ring is of finite dimension it should be obvious, but does anybody know about the situation in more general rings?

REPLY [2 votes]: 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 is contained  in it. That is there is a ring which has no prime with height 1.