A Weaker Notion of Atomicity in Integral Domains

TL;DR

This paper introduces the concept of sub-atomic integral domains, a weaker form of atomicity, and explores their properties, examples, and behavior under various algebraic constructions.

Contribution

It defines sub-atomic domains, studies their characteristics, and demonstrates their independence from classical factorization properties, expanding the taxonomy of integral domains.

Findings

Sub-atomic domains form a class between atomic and non-atomic domains.

The sub-atomic property is preserved under localization and polynomial extensions.

Examples illustrate the independence of sub-atomicity from other factorization properties.

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.