Is there a Noetherian ring which satisfies this property?

Question

Does there exist a Noetherian ring which has arbitrarily large finite strictly increasing sequences of ideals? I know that a Noetherian ring can't have infinite strictly increasing sequences of ideals, by definition, but what about arbitrarily large finite ones?

Answer 1

Sure, even $\mathbb{Z}$ satisfies this. For each $n$:

$(2^{n+1})\subseteq (2^n)\subseteq (2^{n-1})\subseteq...\subseteq (2)$

And all the inclusions are proper.

A more interesting question is can there be a Noetherian ring of infinite dimension, i.e which has aribitrary long increasing finite sequences of prime ideals. This is also possible, see here:

Noetherian ring with infinite Krull dimension (Nagata's example).