On Integral Domains with Prime Divisor Finite Property

TL;DR

This paper introduces and studies tightly prime-divisor-finite domains (TPDF-domains), focusing on their properties and behavior under various algebraic constructions, expanding understanding of prime divisor finiteness in integral domains.

Contribution

It defines TPDF-domains and explores their fundamental properties and how the TPDF property is preserved under localization, $D+M$ constructions, and polynomial extensions.

Findings

TPDF-domains have specific structural properties.

The TPDF property is preserved under localization.

Polynomial rings over TPDF-domains retain the TPDF property.

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.