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!
Best Answer
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 $(x - cy)$.