Hereditary atomicity in integral domains

TL;DR

This paper investigates the properties of hereditarily atomic integral domains, characterizing when certain rings and fields are hereditarily atomic, and exploring implications for polynomial, Laurent polynomial, and power series rings.

Contribution

It provides new characterizations of hereditarily atomic domains, especially in the context of Dedekind domains, polynomial rings, and fields, and addresses open questions about subring properties.

Findings

Certain direct limits of Dedekind domains are Dedekind domains based on atomic overrings

Identifies fields and rings whose polynomial and Laurent polynomial rings are hereditarily atomic

Shows rings of power series are never hereditarily atomic

Abstract

If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.