A characterization of atomicity

TL;DR

This paper investigates the conditions under which atomicity in commutative domains can be characterized, clarifying the relationship between atomicity and the ascending chain condition on principal ideals using monoid and preorder frameworks.

Contribution

It provides a definitive characterization of atomicity in commutative domains, resolving a long-standing open problem with a new ideal-theoretic approach.

Findings

Atomicity characterized via monoids and preorders.

Counterexamples clarify the relationship between atomicity and ACCP.

Resolution of the open problem on atomicity characterization.

Abstract

In [Math. Proc. Cambridge Philos. Soc. 64 (1968), 251-264], P.M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was provided by A. Grams in [Math. Proc. Cambridge Philos. Soc. 75 (1974), 321-329]: Every commutative domain with the ACCP is atomic, but not vice versa. This has led to the question of finding a sensible (ideal-theoretic) characterization of atomicity. The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter-Koch's 2006 monograph on factorization) is still open. We settle it using the language of monoids and preorders.