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The Gaussian integers are well ordered as well. That '<' respects arithmetic operations isn't an aside; it's the core feature.

I did mention that part for a reason. And of course if one accepts the axiom of choice then every set can be well-ordered, but that would not force every ring to be a UFD.

You mentioned it as as an aside. The well ordering property is completely irrelevant.

I appreciate that you're trying to clarify my post, although I did not intend my use of parentheses to imply that the enclosed fact was not important. But I'm unclear on what your ultimate point is. It's certainly not true that well ordering is "completely irrelevant", and I think you must know that if you have the confidence to make such a bold statement, which makes me wonder why you made it in the first place. It plays a very prominent role in many proofs of these facts for the positive integers. It alone is not sufficient to establish unique factorization, but I never claimed it was.

Are you agreeing with me? Disagreeing with me? What sort of response are you expecting? I'd like to have a productive discussion about this, but you're giving me a single bread crumb to go off of here.

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