I recently just completed an exercise which states an atomic domain $A$ is a UFD $\iff$ it satisfies the ascending chain condition on principal ideals. This is all well and good, but because this is an if and only if statement, that seems to suggest there are atomic domains that DON'T satisfy the ACCP. Does anyone know of such an example? I would imagine you need something like the algebraic numbers but I am unsure.
2026-04-04 00:13:28.1775261608
Atomic domain that doesn't satisfy ACCP
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An example of an atomic integral domain not satisfying the ACCP condition is constructed in Section 1 of the 1974 paper Atomic rings and the ascending chain condition for principal ideals by Anne Grams.
(Found via Wikipedia:ACCP.)
I haven't really tried to understand the details yet, so I'll leave it to someone else to explain how it works...