A Weaker Notion of Atomicity in Integral Domains
Mohamed Benelmekki, Brahim Boulayat
TL;DR
The paper introduces a weaker notion of atomicity for integral domains, called sub-atomic domains, where every nonunit divisor of an atomic element is itself atomic. It develops the RIDF framework (restricted-irreducible-divisor-finite domains) and proves that a domain is a U-FFD if and only if it is a sub-atomic RIDF-domain, highlighting a nuanced landscape that lies between atomic and non-atomic domains. The authors analyze these properties under standard constructions such as localization, the $D+M$ construction, and polynomial extensions, and use monoid-domain techniques to construct examples and counterexamples. They show both the independence of sub-atomic and RIDF from classical factorization properties and the ascent behavior of these properties under polynomial extensions in GL- and PSP-domains, contributing a cohesive framework for factorization in non-atomic settings with broad implications for how irreducible divisors control divisibility structures.
Abstract
In classical factorization theory, an integral domain is called \emph{atomic} if every nonzero nonunit element can be written as a finite product of irreducible elements. Here, we introduce and study a weaker notion of atomicity, which relaxes the requirement that all elements admit a factorization into irreducibles. Namely, we say that an integral domain is \emph{sub-atomic} if every nonunit divisor of an atomic element is also atomic. We further consider several factorization properties associated with this notion. Then, we investigate the basic properties of such domains, provide examples, and explore the behavior of the sub-atomic property under standard constructions such as localization, polynomial rings, and $D+M$ constructions. Our results highlight the independence of the sub-atomic property from other classical factorization properties and introduce an important class of integral domains that lies between atomic and non-atomic domains.