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Noncatenary Unique Factorization Domains

Alexandra Bonat, S. Loepp

Abstract

We demonstrate a class of local (Noetherian) unique factorization domains (UFDs) that are noncatenary at infinitely many places. In particular, if $A$ is in our class of UFDs, then the prime spectrum of $A$ contains infinitely many disjoint (except at the maximal ideal) noncatenary subsets. As a consequence of our result, there are infinitely many height one prime ideals $P$ of $A$ such that $A/P$ is not catenary. We also construct a countable local UFD $A$ satisfying the property that for every height one prime ideal $P$ of $A$, $A/P$ is not catenary.

Noncatenary Unique Factorization Domains

Abstract

We demonstrate a class of local (Noetherian) unique factorization domains (UFDs) that are noncatenary at infinitely many places. In particular, if is in our class of UFDs, then the prime spectrum of contains infinitely many disjoint (except at the maximal ideal) noncatenary subsets. As a consequence of our result, there are infinitely many height one prime ideals of such that is not catenary. We also construct a countable local UFD satisfying the property that for every height one prime ideal of , is not catenary.
Paper Structure (4 sections, 17 theorems, 9 equations, 2 figures)

This paper contains 4 sections, 17 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Main Result
  4. A Very Noncatenary UFD

Key Result

Lemma 2.2

Let $T$ be a complete local ring with maximal ideal $M$, let $C$ be a countable set of prime ideals in $\text{Spec}\, T$ such that $M \notin C$, and let $D$ be a countable set of elements of $T$. If $I$ is an ideal of $T$ which is contained in no single $P$ in $C$, then $I \not \subset \bigcup \{(P

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (32)

  • Definition 2.1
  • Lemma 2.2: heitmann, Lemma 2
  • Definition 2.3: heitmann
  • Lemma 2.4: loepp, Lemma 11
  • Lemma 2.5
  • proof
  • Definition 2.6: heitmann
  • Lemma 2.7: heitmann, Lemma 6
  • Lemma 2.8
  • proof
  • ...and 22 more