Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains

Abstract

In this article we survey recent results concerning the set of prime ideals in two-dimensional Noetherian integral domains of polynomials and power series. We include a new result that is related to current work of the authors [Celikbas et al., Prime Ideals in Quotients of Mixed Polynomial-Power Series Rings; see http://www.math.unl.edu/~swiegand1 (preprint)]: Theorem 5.4 gives a general description of the prime spectra of the rings \(R[\![x,y]\!]/P,R[\![x]\!][y]/Q\) and R[y][​[x]​]∕Q′, where x and y are indeterminates over a one-dimensional Noetherian integral domain R and P, Q, and Q′ are height-one prime ideals of R[​[x, y]​], R[​[x]​][y], and R[y][​[x]​], respectively. We also include in this survey recent results of Eubanks-Turner, Luckas, and Saydam describing prime spectra of simple birational extensions R[​[x]​][f(x)∕g(x)] of R[​[x]​], where f(x) and g(x) are power series in R[​[x]​] such that f(x) ≠ 0 and is a prime ideal of R[​[x]​][y]—this is a special case of Theorem 5.4. We give some examples of prime spectra of homomorphic images of mixed power series rings when the coefficient ring R is the ring of integers \(\mathbb{Z}\) or a Henselian domain.