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On Generalized Lengths of Factorizations in Dedekind and Krull Domains

  • Scott T. Chapman
  • Michael Freeze
  • William W. Smith
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has con­centrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

Keywords

Prime Ideal Class Group Integral Domain Factorization Property Ideal Class 
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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Scott T. Chapman
    • 1
  • Michael Freeze
    • 2
  • William W. Smith
    • 2
  1. 1.Department of MathematicsTrinity UniversitySan AntonioUSA
  2. 2.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

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