A Weaker Notion of Atomicity in Integral Domains

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.

💡 Research Summary

The paper introduces a new, weaker notion of atomicity in integral domains, called sub‑atomicity, and investigates its basic properties, examples, and behavior under standard constructions. Classical factorization theory defines an integral domain to be atomic if every non‑zero non‑unit can be expressed as a finite product of irreducibles (atoms). While this condition is natural, many important non‑Noetherian domains fail to be atomic because they contain elements that admit no factorization into atoms. Existing “weak atomicity” concepts—near‑atomic, almost‑atomic, and quasi‑atomic—relax the requirement that each element itself factor into atoms; instead they demand that each non‑unit divide some atomic element. However, all these notions still presuppose the existence of at least one atom in the domain.

The authors propose a still weaker condition that does not require any atoms at all. An integral domain (D) is called sub‑atomic if every non‑unit divisor of an atomic element of (D) is itself atomic. In other words, whenever an element (a) can be written as a product of atoms, any non‑unit factor (b) of (a) must also admit a factorization into atoms. This definition automatically includes all atomic domains (since every element is atomic, the condition is vacuous) and also includes “antimatter” domains—domains with no atoms whatsoever—because the condition is trivially satisfied when there are no atomic elements. Thus sub‑atomicity sits strictly between atomicity and the complete absence of atoms.

The paper first reviews the classical hierarchy: \