ABSTRACT
We present examples of Noetherian and non-Noetherian integral domains which can be built inside power series rings. Given a power series ring R* over a Noetherian integral domain R and given a subfield L of the total quotient ring of R* with R ⊆ L, we construct subrings A and B of L such that B is a localization of a nested union of polynomial rings over R and B ⊆ A := L ∩ R*. We show in certain cases that flatness of a related map on polynomial rings is equivalent to the Noetherian property for B. Moreover if B is Noetherian, then B = A. We use this construction to obtain for each positive integer n an explicit example of a 3-dimensional quasilocal unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely n prime ideals of height two, and each prime ideal of B of height two is not finitely generated.
