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

## TL;DR

This paper constructs specific local unique factorization domains with multiple disjoint maximal chains of prime ideals of prescribed lengths, revealing complex prime ideal structures possible in such domains.

## Contribution

It introduces a method to create UFDs with multiple disjoint prime ideal chains of specified lengths, expanding understanding of prime ideal configurations.

## Key findings

- Existence of UFDs with disjoint prime chains of arbitrary lengths
- Construction of local Noetherian UFDs with prescribed prime ideal structures
- Demonstration of unusual prime ideal configurations in UFDs

## Abstract

We show that, given integers $n_1,n_2, \ldots ,n_k$ 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 $n_1, n_2, \ldots ,n_k$ 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.