Noetherian domains with the same prime ideal structure as Z{2)‎[x]

Abstract EN

Which two-dimensional Noetherian domains A have the same prime ideal structure as the ring Z(2)[x] of polynomials over the rational numbers with odd denominators? A significant property of Z(2)[x] is this: For every finite set raj,...,!!!,, of height-two maximal ideals of Z(2)[jc], there are infinitely many height-one prime ideals which are contained in each m- but not contained in any other maximal ideal.

In this article we show that this property holds for every domain B of the form B := R[x][glf], where/? is a one-dimensional semilocal non-Henselian Hilbertian domain,/ g is an /?[x]-sequence, and//?[x] n R = (0).

In view of previous results of William Heinzer, David Lantz, and Wiegand, this result implies that, if R is countable, then the prime ideal structure of B is the same as that of Z(2)[x] if and only if B has exactly one non-maximal height-one prime ideal which is contained in infinitely many maximal ideals.

Equivalently, the domain R has a unique maximal ideal m and the images/ and g in (/?/m)[x] of/and g are relatively prime.