## Journal of Commutative Algebra

### Examples of non-Noetherian domains inside power series rings

#### 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

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

#### 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

#### References

• S.S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348.
• M. Atiyah and I. Macdonald, Introduction to commutative algebra, Addison-Wesley, London, 1969.
• J.W. Brewer and E.A. Rutter, $D+M$ constructions with general overrings, Michigan Math. J. 23 (1976), 33-42.
• I.S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106.
• J. David, A characteristic zero non-Noetherian factorial ring of dimension three, Trans. Amer. Math Soc. 180 (1973), 315-325.
• E. Evans, A generalization of Zariski's main theorem, Proc. Amer. Math. Soc. 26 (1970), 45-48.
• S. Gabelli and E. Houston, Coherentlike conditions in pullbacks, Michigan Math. J. 44 (1997), 99-123.
• R. Gilmer, Prüfer-like conditions on the set of overrings of an integral domain, Lect. Notes Math. 311, Springer, Berlin, 1973.
• –––, Multiplicative ideal theory, Queen's Papers Pure Appl. Math. 90, Queen's University, Kingston, ON, 1992.
• A. Grothendieck, Élement de Géométrie Algébrique IV, Publ. Math. Inst. Haut. Étud. Sci. 24, 1965.
• W. Heinzer and M. Roitman, The homogeneous spectrum of a graded commutative ring, Proc. Amer. Math. Soc. 130 (2001), 1573-1580.
• W. Heinzer, C. Rotthaus and S. Wiegand, Building Noetherian domains inside an ideal-adic completion II, in Advances in commutative ring theory Lect. Notes Pure Appl. Math. 205, Dekker, New York, 1999.
• –––, Noetherian rings between a semilocal domain and its completion, J. Algebra 198 (1997), 627-655.
• –––, Building Noetherian domains inside an ideal-adic completion, in Abelian groups, module theory, and topology, Dikran Dikranjan and Luigi Salce, eds., Lect. Notes Pure Appl. Math. 201, Dekker, New York, 1998.
• –––, Noetherian domains inside a homomorphic image of a completion, J. Alg. 215 (1999), 666-681.
• –––, Examples of integral domains inside power series rings, in Commutative rings and applications, Lect. Notes Pure Appl. Math. 231, Dekker, New York, 2003.
• –––, Non-finitely generated prime ideals in subrings of power series rings, in Rings, modules, algebras, and abelian groups, Lect. Notes Pure Appl. Math. 236, Dekker, New York, 2004.
• –––, Power series over Noetherian rings, in progress.
• W. Heinzer and J. Sally, Extensions of valuations to the completion of a local domain, J. Pure Appl. Alg. 71 (1991), 175-185.
• I. Kaplansky, Commutative rings, Allyn Bacon, Boston, 1970.
• G. Leuschke and R. Wiegand, Cohen-Macaulay representations, Math. Surv. Mono. 181, American Mathematical Society, Providence, RI, 2012.
• H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1989.
• –––, Commutative algebra, second edition, Benjamin, New York, 1980.
• M. Nagata, Local rings, John Wiley & Sons, New York, 1962.
• C. Peskine, Une généralsation du “main theorem” de Zariski, Bull. Sci. Math. 90 (1966), 377-408.
• C. Rotthaus, Nicht ausgezeichnete, universell japanische Ringe, Math. Z. 152 (1977), 107-125.
• C. Rotthaus and L. Sega, On a class of coherent regular rings, Proc. Amer. Math. Soc. 135 (2007), 1631-1640.
• P. Samuel, Unique factorization domains, Tata Institute of Fundamental Research, Bombay, 1964.
• P. Valabrega, On two-dimensional regular local rings and a lifting problem, Ann. Scuola Norm. Sup. Pisa 27 (1973), 1-21.
• O. Zariski and P. Samuel, Commutative algebra II, Van Nostrand, Princeton, 1960. \noindentstyle