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On Integral Domains with Prime Divisor Finite Property

Mohamed Benelmekki

Abstract

An integral domain $D$ is called a \emph{prime-divisor-finite domain} (PDF-domain) if every nonzero element has only finitely many nonassociate prime divisors. A domain $D$ is said to be a \emph{tightly prime-divisor-finite domain} (TPDF-domain) if it is a PDF-domain and every nonzero nonunit element admits at least one prime divisor. In this paper, we study TPDF-domains. We investigate some basic properties of these domains and examine the behavior of the TPDF property under standard constructions such as localization, $D+M$ constructions, and polynomial rings.

On Integral Domains with Prime Divisor Finite Property

Abstract

An integral domain is called a \emph{prime-divisor-finite domain} (PDF-domain) if every nonzero element has only finitely many nonassociate prime divisors. A domain is said to be a \emph{tightly prime-divisor-finite domain} (TPDF-domain) if it is a PDF-domain and every nonzero nonunit element admits at least one prime divisor. In this paper, we study TPDF-domains. We investigate some basic properties of these domains and examine the behavior of the TPDF property under standard constructions such as localization, constructions, and polynomial rings.
Paper Structure (8 sections, 17 theorems, 4 equations)

This paper contains 8 sections, 17 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Background
  3. General Notation
  4. Factorization in integral domains
  5. Content of Polynomials and PSP-Domains
  6. Monoid domains
  7. Strong Furstenberg PDF-domains
  8. The $D+M$ construction and localisation

Key Result

Proposition 3.1

An integral domain $D$ is a strong Furstenberg domain if and only if it is a Furstenberg AP-domain.

Theorems & Definitions (37)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • Corollary 3.6
  • ...and 27 more