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I am looking for a commutative ring R with identity, with infinitely many ideals, yet every proper ideal contains (as a subset) only finitely many ideals of R. I know such a ring cannot be Artin (because Artin rings have only a finite number of max ideals, so one of them contains infinitely many proper ideals) I also know Artin implies either Not Noetherian or nozero Krull dimension. Does such a ring exist? I'm stuck here.

Wdunn
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Such a ring is trivially Artinian: any descending chain of ideals must terminate, since as soon as you reach a proper ideal it has only finitely many subideals. But, as you observe, such a ring cannot be Artinian. Thus no such ring exists.

Eric Wofsey
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  • Thank you Eric Wofsey! This answer was so close to me, I completely overlooked it! Thanks again! – Wdunn Sep 17 '20 at 21:14