A characterization of atomicity

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.