Noetherian Ring with Infinite Krull Dimension
This one is due to Nagata. Consider a polynomial ring in countably infinitely many indeterminates over a field Let, for each , denote the prime ideal . Then, define the (multiplicative) subset . The localization will be our counterexample.
We claim that is Noetherian. To see this, consider a prime ideal . First, each is maximal and each is contained in only finitely many . So, is contained in only finitely many . For each , is finitely generated. Thus, we can obtain a finite list of elements of that generate each Note that, by Hilbert’s basis theorem, is Noetherian. Let Observe that for all but finitely many maximal ideals but the are not any of the in which is contained. Moreover, for all . Therefore, gives in each localization and so is finitely generated. As all prime ideals of are finitely generated, it is Noetherian.
Lastly, for each there is a chain of prime ideals. The supremum then of the lengths of chains of prime ideals in is infinite e.g. has infinite Krull dimension.