# Is this ring Noetherian? Artinian?

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...

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 $$0$$.