Commutative Algebra pp 225-238 | Cite as

# How to Construct Huge Chains of Prime Ideals in Power Series Rings

## Abstract

Let *R* be a commutative ring with identity. It is well known that if each chain of prime ideals in *R* has length at most *n*, then each chain of prime ideals in the polynomial ring *R*[*X*] has length at most 2*n* + 1. For the power series ring *R*[[*X*]], there is however no similar upper bound on lengths of chains of its prime ideals. In fact, in some special cases, there may exist chains of prime ideals in *R*[[*X*]] with huge lengths (e.g., \(2^{\aleph _{1}}\)) even if each chain of prime ideals in *R* has length at most one. The purpose of this work is to give a brief review on known constructions of chains of prime ideals in *R*[[*X*]] in those cases. By taking into account the techniques which are used in the constructions and possibly by applying some new tools, we hope to construct huge chains of prime ideals in *R*[[*X*]] in more general cases.

## Keywords

Almost Dedekind domain Krull dimension Non-SFT ring Power series ring Valuation domain## Mathematics Subject Classification (2010):

13A15 13C15 13F05 13F25 13F30.## References

- 1.J.T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc.
**177**, 299–304 (1973)CrossRefzbMATHMathSciNetGoogle Scholar - 2.J.T. Arnold, Power series rings over Prüfer domains, Pacific J. Math.
**44**, 1–11 (1973)CrossRefzbMATHMathSciNetGoogle Scholar - 3.J. Coykendall, The SFT property does not imply finite dimension for power series rings. J. Algebra
**256**(1), 85–96 (2002)CrossRefzbMATHMathSciNetGoogle Scholar - 4.J. Coykendall, in
*Progress on the Dimension Question for Power Series Rings*, ed. by J.W. Brewer, S. Glaz, W.J. Heinzer, B.M. Olberding, Multiplicative Ideal Theory in Commutative Algebra, (Springer, New York, 2006) pp. 123–135Google Scholar - 5.P. Eakin, A. Sathaye,
*Some Questions about the Ring of Formal Power Series*, in: Commutative Algebra (Fairfax Va., 1979), Lecture Notes in Pure and Appl. Math. vol. 68, (Dekker, New York, 1982) pp. 275–286Google Scholar - 6.R. Gilmer,
*Multiplicative Ideal Theory*(Marcel Dekker, New York, 1972)zbMATHGoogle Scholar - 7.M. Henriksen, On the prime ideals of the ring of entire functions. Pacific J. Math.
**3**, 711–720 (1953)CrossRefzbMATHMathSciNetGoogle Scholar - 8.B.G. Kang, M.H. Park, A localization of a power series ring over a valuation domain. J. Pure Appl. Algebra
**140**(2), 107–124 (1999)CrossRefzbMATHMathSciNetGoogle Scholar - 9.B.G. Kang, M.H. Park,
*Krull Dimension of Mixed Extensions*. J. Pure Appl. Algebra**213**(10), 1911–1915 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 10.B.G. Kang, M.H. Park, Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable. J. Algebra
**378**, 12–21 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 11.B.G. Kang, K.A. Loper, T.G. Lucas, M.H. Park, P.T. Toan,
*The Krull dimension of power series rings over non-SFT rings*. J. Pure Appl. Algebra**217**(2), 254–258 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 12.K.A. Loper, T.G. Lucas, Constructing chains of primes in power series rings. J. Algebra
**334**, 175–194 (2011)CrossRefzbMATHMathSciNetGoogle Scholar - 13.K.A. Loper, T.G. Lucas, Constructing chains of primes in power series rings II. J. Algebra Appl.
**12**(1), 1250123, 30 (2013)Google Scholar - 14.A. Seidenberg, A note on the dimension theory of rings. Pacific J. Math.
**3**, 505–512 (1953)CrossRefzbMATHMathSciNetGoogle Scholar - 15.A. Seidenberg, On the dimension theory of rings II. Pacific J. Math.
**4**, 603–614 (1954)CrossRefzbMATHMathSciNetGoogle Scholar