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Open Problems in Commutative Ring Theory

Abstract

This chapter consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

Keywords

  • Prüfer ring
  • Homological dimensions
  • Integral closure
  • Group ring
  • Grade
  • Complete ring
  • McCoy ring
  • Straight domain
  • Divided domain
  • Integer-valued polynomials
  • Factorial
  • Density
  • Matrix ring
  • Overring
  • Absorbing ideal
  • Kronecker function ring
  • Stable ring
  • Divisorial domain
  • Mori domain
  • Finite character
  • PvMD
  • Semistar operation
  • Star operation
  • Jaffard domain
  • Locally tame domain
  • Factorization
  • Spectrum of a ring
  • Integral closure of an ideal
  • Rees algebra
  • Rees valuation

Mathematics Subject Classification (2010):

  • 13-02
  • 13A05
  • 13A15
  • 13A18
  • 13B22
  • 13C15
  • 13D05
  • 13D99
  • 13E05
  • 13F05
  • 13F20
  • 13F30
  • 13G05

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Acknowledgements

We thank all the commutative algebraists who contributed open problems to this chapter. The list of contributors is as follows: D.D. Anderson (Problem 8), A. Badawi (Problem 30), P.-J. Cahen (Problems 14 and 15), J.-L. Chabert (Problems 16–18), J. Elliott (Problems 19–23), C.A. Finocchiaro and M. Fontana (Problem 36), S. Frisch (Problems 28 and 39), S. Gabelli (Problems 32 and 33), A. Geroldinger (Problem 38), S. Glaz (Problems 1–3), L. Hummel (Problem 7), K. Johnson (Problems 24 and 25), S. Kabbaj (Problem 37), T.G. Lucas (Problems 9–12), B. Olberding (Problems 29 and 31), G. Peruginelli (Problem 26), G. Picavet and M. Picavet-L’Hermitte (Problem 13), R. Schwarz (Problems 4–6), I. Swanson (Problems 41–44), N.J. Werner (Problems 27 and 28), S. Wiegand and R. Wiegand (Problem 40), M. Zafrullah (Problems 34 and 35).

Note that while this chapter and reference [54] were in proof, Problem 32a has been answered negatively. A counter example is given in [54, Example 3.9] with a 2-dimensional Prüfer domain.

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Cahen, PJ., Fontana, M., Frisch, S., Glaz, S. (2014). Open Problems in Commutative Ring Theory. In: Fontana, M., Frisch, S., Glaz, S. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0925-4_20

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