# Commutative Artinian ring with infinitely many ideals.

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!

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

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 $$(x - cy)$$.
• @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. Feb 4, 2021 at 12:40