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Can anyone give me an example of one? I know it's equivalent to finding a noetherian that isn't PID, but I'm not sure!

the man
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  • $\mathbb{C}[x,y]$ – Levent Jan 25 '18 at 12:58
  • @Levent Can you explain why please? – the man Jan 25 '18 at 13:04
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    i dont think the phrase "i know it's equivalent to finding a noetherian that isnt a pid" is correct (see the answer below). maybe in this way you could get the question open again. for instance you could ask: "a example of noetherian ring that is a pid and not euclidean" (check that this is not already asked also) – Jacques Saliba Apr 09 '18 at 15:24

1 Answers1

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The ring of integers of $\mathbb{Q}(\sqrt{-19})$ is a principal ideal domain that is not Euclidean.

lhf
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  • I think some people found this answer more sophisticated than necessary. – user26857 Jan 25 '18 at 16:29
  • @user26857, perhaps, but this example makes a strong point: A ring may be very Noetherian (a PID!) but not Euclidean. – lhf Jan 25 '18 at 17:20