Maximal Chains of Prime Ideals of Different Lengths in Unique Factorization Domains

Abstract

We show that, given integers $n_1,n_2,\dots ,n_k$ with $2<n_1<n_2<\cdots <n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1,n_2,\dots ,n_k$ which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.

20 pages, 3 figures. To be published in the Rocky Mountain Journal of Mathematics