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UFD imply ACCP (ascending chain condition for principal ideals). But is it necessarily UFD implies ACC also (ascending chain condition for ideals)?


marked as duplicate by user26857, 6005, Mark Bennet, user147263, mrf Sep 22 '15 at 22:17

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  • $\begingroup$ If it were so, you'd probably have seen a theorem stating it. So guess that there are UFDs that aren't Noetherian. Where you should first look for (counter)examples depends on your background. $\endgroup$ – Daniel Fischer Sep 22 '15 at 11:30
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    $\begingroup$ Is there a reason for deleting the previous question after someone answered one of your questions and after I gave you a link where you could find the example given below? $\endgroup$ – user26857 Sep 22 '15 at 20:51

No. If $K$ is a field, $K[X_1,X_2,\dots, X_n,\dots]$ is an example of a non-noetherian UFD.


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