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let $D$ be an integral domain from now on.

In the lecture, my professor proves TFAE:
1. $D$ is a UFD
2. $D$ satisfies ACCP and for any $a$ in $D$ we have $a$ is prime $\iff$ $a$ is irreducible

On the other hand, the GCD domain article in Wikipedia states that TFAE:
1. $D$ is a UFD
2. $D$ is a GCD domain and $D$ satisfies ACCP

Furthermore, the MO question Integral domain is ufd iff atomic and gcd domain says TFAE:
1. $D$ is a UFD
2. $D$ is a GCD domain and $D$ is a factorization domain (aka atomic domain)

Now here comes a problem, the Atomic domain article in Wikipedia claims that $D$ satisfies ACCP $\implies$ $D$ is a factorization domain but the converse is not true. So something must be wrong here, right? Being factorization domain is slightly weaker than having ACCP according to the Atomic domain article. So either the Atomic domain article is wrong or one of the Wikipedia or MO characterization is wrong. Which of the above are actually correct, what do you think?

Alex Vong
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1 Answers1

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Nothing is wrong. ACCP and factorization domain are not equivalent in general, but they are equivalent for GCD domains.

Eric Wofsey
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  • Thank you, I will try to prove it as an exercise then. – Alex Vong Dec 02 '16 at 05:20
  • @EricWofsey Can you give a reference for a proof of GCD + atomic $\Rightarrow$ ACCP? Google search is being futile here. – Atom Apr 04 '23 at 16:11
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    @Atom: It's probably easier to first prove unique factorization (see https://math.stackexchange.com/questions/1595010/prove-that-in-any-gcd-domain-every-irreducible-element-is-prime for instance), from which ACCP follows. – Eric Wofsey Apr 04 '23 at 18:10
  • That did it for me! Thanks. – Atom Apr 04 '23 at 18:25