**Background**: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. Without this assumption, there are counter-examples.


Thanks.



Does every regular Noetherian domain have finite Krull dimension?
No. An example is given in K. Fujita, [*Infinite dimensional Noetherian Hilbert domains,*](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.hmj/1206136627&page=record) Hiroshima Math. J. 5 (1975), 181-185.


