Exhibit a non-principal ideal

Question

It is well-known that in a Noetherian UFD, every height one prime ideal is principal. I was wondering whether this statement holds if one replaces 'UFD' with 'locally factorial domain'. I am aware of the fact that 'locally factorial' in general does not imply 'factorial', i.e., 'unique factorization' (though have never come accross locally factorial domains that are not UFDs), so I believe that there is a counterexample to this. However I cannot come up with any.

Answer 1

A Dedekind domain is locally factorial in that for each (nonzero) prime ideal $P$ we have $D_{P}$ a UFD. Now $\mathbb Z[\sqrt{-5}]$ is a Dedekind domain. Check that the ideal $(2,1+\sqrt{-5})$ is a prime ideal.

Also, Lemma 3.1 in this paper [CDZ] says:

An integral domain $D$ is factorial if and only if every minimal prime of a nonzero principal ideal of $D$ is principal.

Now a Dedekind domain $D$ is an integrally closed noetherian domain of Krull dimension $1.$ So every nonzero prime ideal of a Dedekind domain is the minimal prime of a principal ideal, being of height one. By Lemma 3.1 of [CDZ], if every nonzero prime ideal of a Dedekind domain $D$ is principal, then $D$ is a PID. So take a Dedekind domain such as $\mathbb Z[\sqrt{-5}]$ which know to be not a PID (i.e. not a UFD). Then you know, without an exhibition that there must be at least one nonzero prime ideal $P$ that is not principal.

[CDZ] Gyu Whan Chang, Tiberiu Dumitrescu and Muhammad Zafrullah, Locally GCD domains and the ring $D+XD_{S}[X]$, Bull. Iranian Math. Soc. 42(2016), 263-284.