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This question already has an answer here:

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

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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
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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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