On near atomicity and a characterization of the FF property

Abstract

A commutative cancellative monoid is atomic if every nonunit factors into atoms, and an integral domain is atomic if its multiplicative monoid of nonzero elements is atomic. Several weakenings of atomicity have been introduced and studied during the past decade, including near atomicity, almost atomicity, and quasi-atomicity. Although nearly atomic monoids that are not atomic were already known, whether there exist nearly atomic integral domains that are not atomic had remained open. We answer this question affirmatively by constructing an explicit nearly atomic integral domain that is not atomic. We also strengthen the classical Anderson--Anderson--Zafrullah characterization of the finite factorization property by proving that an integral domain is an FFD if and only if it is both nearly atomic and IDF. We conclude by showing that near atomicity cannot be weakened to almost atomicity in this characterization, even within the class of IDF domains.