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Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains

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Commutative Algebra

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.

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Notes

  1. 1.

    For the definition of “excellent ring” see [15, p. 260]. Basically “excellence” means the ring is catenary and has other nice properties that polynomial rings over a field possess.

  2. 2.

    Essentially a “Henselian” ring is one that satisfies Hensel’s Lemma; see the definition in [20].

  3. 3.

    Since axiom II 0 holds, this axiom could be stated here without saying “v is not maximal.”

  4. 4.

    The term “Le (T)” is defined in Notation 2.1.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Ela Celikbas

  2. Department of Mathematics, Loyola Marymount University, Los Angeles, CA, 90045, USA

    Christina Eubanks-Turner

  3. Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, 68588, USA

    Sylvia Wiegand

Authors
  1. Ela Celikbas
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  2. Christina Eubanks-Turner
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  3. Sylvia Wiegand
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Corresponding author

Correspondence to Sylvia Wiegand .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  2. Department of Mathematics, Graz University of Technology, Graz, Austria

    Sophie Frisch

  3. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Sarah Glaz

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Celikbas, E., Eubanks-Turner, C., Wiegand, S. (2014). Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains. In: Fontana, M., Frisch, S., Glaz, S. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0925-4_4

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