If R is a Noetherian ring, is $R[X]/((X-1)^2X)$ Noetherian? Artinian? So first I have to understand if it is Noetherian or Artinian, and then prove it or find chains of ideals that don't stabilize. So often I am a bit lost because here it is not even clear to me if these things are Noetherian or Artinian in the first place. I know that the ideals from the chains will be in the form $J/((X-1)^2X)$ where $J$ contains $((X-1)^2X)$, but then...


It is noetherian, because it's a quotient of a noetherian ring.

But it is not Artinian in general. For instance if $R=\Bbb Z$ there's a chain of prime ideals $(2,\bar X)\supset (\bar X)$ so that $\dim R[X]/((X−1)^2X) >0$, whereas Artin rings have Krull dimension $0$.

  • $\begingroup$ For Noetherian part, it is clear, but is it possible to see this without the help of Krull dimension, which we didn't see yet? $\endgroup$ – roi_saumon Oct 28 '18 at 21:53
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    $\begingroup$ @roi_saumon: well, without invoking explicitely the Krull dimension, you should nonetheless know that in an Artin ring all prime ideals are maximal. $\endgroup$ – Andrea Mori Oct 28 '18 at 22:14

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