Maximal chains of prime ideals of different lengths in unique factorization domains

Abstract

We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 \lt n_1 \lt n_2 \lt \cdots \lt n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$ 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.