Chapter

Commutative Algebra

pp 55-82

Date:

Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains

  • Ela CelikbasAffiliated withDepartment of Mathematics, University of Missouri
  • , Christina Eubanks-TurnerAffiliated withDepartment of Mathematics, Loyola Marymount University
  • , Sylvia WiegandAffiliated withDepartment of Mathematics, University of Nebraska–Lincoln Email author 

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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.

Keywords

Commutative ring Noetherian ring Integral domain Polynomial ring Power series ring Prime ideals Prime spectrum

1991 Mathematics Subject Classification

Primary 13B35 13J10 13A15