In [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, A. Grams [Math. Proc. Cambridge Phil. Soc. 75 (1974), No. 3, 321-329] showed (by way of a counterexample) that Cohn's statement is wrong: Every commutative domain satisfying the ACCP is atomic, but not the other way around. It is therefore natural to wonder if Cohn's claim can be, in a sense, fixed by providing a sensible characterization (say, of a somewhat ideal-theoretic flavour) of when a commutative domain (or, more generally, a cancellative commutative monoid) is atomic.

Q. I was confirmed that the question is open and well known (at least in some circles), but haven't found any solid evidence in support of this. Do you happen to know of any papers, conference proceedings, etc. (the older, the better) where the question, although somewhat vague in its formulation, is explicitly stated?

Let me recall that a (multiplicatively written) monoid is atomic if every non-unit factors as a product of atoms (i.e., elements that cannot be written as a product of two non-units); and a domain is atomic if so is the multiplicative monoid of its non-zero elements.