Atomicity in integral domains

Abstract

In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and, therefore, divisibility by and factorizations into atoms are not possible. In the other (more interesting) side of the spectrum, we find the atomic algebraic structures, where essentially every element factors into atoms (the study of such objects is known as factorization theory). In this paper, we survey some of the most fundamental results on the atomicity of cancellative commutative monoids and integral domains, putting our emphasis on the latter. We mostly consider the realm of atomic domains. For integral domains, the distinction between being atomic and satisfying the ascending chain condition on principal ideals, or ACCP for short (which is a stronger and better-behaved algebraic condition) is subtle, so atomicity has been often studied in connection with the ACCP: we consider this connection at many parts of this survey. We discuss atomicity under various classical algebraic constructions, including localization, polynomial extensions, $D+M$ constructions, and monoid algebras. Integral domains having all their subrings atomic are also discussed. In the last section, we explore the middle ground of the spectrum of atomicity: some integral domains where some of but not all the elements factor into atoms, which are called quasi-atomic and almost atomic. We conclude providing techniques from homological algebra to measure how far quasi-atomic and almost atomic domains are from being atomic.