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Intro: An immediate example of Artinian ring is any ring with finitely many ideals. Recently, when I was thinking about Artinian rings, I realized that this is actually the only example I really know. I always implicitly assumed there are lots of Artin rings with infinitely many ideals, but I can't really find such a ring.

Question: Please, can you point out an example of unital, commutative Artinian ring with infinitely many ideals?

Non-commutative example is welcomed too, but commutative one is preferred.

My thoughts (here I consider only commutative case) Usually when I think about some concrete ring, it is an integral domain. But only Artin domain is a field so this is not a good way to go.

I've read that Artinian ring is a finite product of Artin local ring. So it seems to be a good idea to look at local Artin rings with the maximal ideal that is not principal. However, I am still not really used to localizations (good enough to understand some proof but not very experienced with actually using this technique) so I can't use this to actually produce a concrete example.

Your thoughts and examples are really appreciated. Thanks!

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  • $\begingroup$ Hint: Take $k[x,y] / (x^2,xy, y^2)$ $\endgroup$ Feb 4, 2021 at 12:23

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Take your favourite infinite field $k$ and consider the Artinian ring $k[x,y] / (x^2,xy,y^2)$. Then for any $c$ in $k$, there's an ideal $(x - cy)$.

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    $\begingroup$ Is there some geometric intuition behind this? $\endgroup$
    – Claudius
    Feb 4, 2021 at 12:38
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    $\begingroup$ @Claudius You can think of $\operatorname{Spec} k[x,y] / (x^2,xy,y^2)$ as a point in $2$-space fattened up in all directions. Its underlying space is still $0$-dimensional, as it should be with Artinian rings. The closed subschemes corresponding to $(x - cy)$ are then 'lines' with various (infinitely many) slopes. $\endgroup$ Feb 4, 2021 at 12:40
  • $\begingroup$ Thanks for the answer. This was something I was searching for. $\endgroup$
    – dmk
    Feb 4, 2021 at 12:45
  • $\begingroup$ Wow, so many intriguing things to ask about! (And they're probably insufferably trivial to you, so I hope nothing I ask annoys you, but I feel like what you said has a wealth of unlearned intuition for me.) So the spectrum is "thin but broad." Considering that it's a singleton, it's weird that there is information contained in the point (the subschemes, that I am not familiar with thinking about.). Is this related to thinking of the spectrum as a manifold, and the subschemes are related to the dimension of the tangent space? $\endgroup$
    – rschwieb
    Feb 4, 2021 at 15:02
  • $\begingroup$ @rschwieb Don't worry, I'm happy to expand on it. :) It's a bit tough to cram it in a comment, but the 'fattening' corresponds more or less to nilpotent elements in your ring. When I first learned AG I dismissed the continual talk of 'fat point this' and 'fuzziness that' as vague nonsense but I've come to learn that it can be really helpful. The intuition behind the example I gave happens to be worked out in Section 4.2 of Vakil's FoAG, which is specifically about the geometric intuition of nilpotents. $\endgroup$ Feb 4, 2021 at 19:34

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