Journal of Commutative Algebra

Examples of non-Noetherian domains inside power series rings

William Heinzer, Christel Rotthaus, and Sylvia Wiegand

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Abstract

Given a power series ring $R^*$ over a Noetherian integral domain $R$ and an intermediate field $L$ between $R$ and the total quotient ring of $R^*$, the integral domain $A = L \cap R^*$ often (but not always) inherits nice properties from $R^*$ such as the Noetherian property. For certain fields $L$ it is possible to approximate $A$ using a localization $B$ of a particular nested union of polynomial rings over $R$ associated to $A$; if $B$ is Noetherian, then $B = A$. If $B$ is not Noetherian, we can sometimes identify the prime ideals of $B$ that are not finitely generated. We have obtained in this way, for each positive integer $m$, a three-dimensional local unique factorization domain $B$ such that the maximal ideal of $B$ is two-generated, $B$ has precisely $m$ prime ideals of height~2, each prime ideal of $B$ of height~2 is not finitely generated and all the other prime ideals of $B$ are finitely generated. We examine the structure of the map $\text{Spec\,} A \to \text{Spec\,} B$ for this example. We also present a generalization of this example to dimension four. This four-dimensional, non-Noetherian local unique factorization domain has exactly one prime ideal $Q$ of height three, and $Q$ is not finitely generated.

Article information

Source
J. Commut. Algebra, Volume 6, Number 1 (2014), 53-93.

Dates
First available in Project Euclid: 2 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1401715578

Digital Object Identifier
doi:10.1216/JCA-2014-6-1-53

Mathematical Reviews number (MathSciNet)
MR3215561

Zentralblatt MATH identifier
1304.13003

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory 13B35: Completion [See also 13J10] 13J10: Complete rings, completion [See also 13B35]

Keywords
Power series Noetherian and non-Noetherian integral domains

Citation

Heinzer, William; Rotthaus, Christel; Wiegand, Sylvia. Examples of non-Noetherian domains inside power series rings. J. Commut. Algebra 6 (2014), no. 1, 53--93. doi:10.1216/JCA-2014-6-1-53. https://projecteuclid.org/euclid.jca/1401715578


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