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Non-Noetherian Commutative Ring Theory pp 117–137Cite as

On Generalized Lengths of Factorizations in Dedekind and Krull Domains

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Part of the book series: Mathematics and Its Applications ((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.

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Authors and Affiliations

  1. Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, Texas, 78212-7200, USA

    Scott T. Chapman

  2. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599-3250, USA

    Michael Freeze & William W. Smith

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  1. Scott T. Chapman
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  2. Michael Freeze
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  3. William W. Smith
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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Scott T. Chapman

  2. Department of Mathematics, The University of Connecticut, Storrs, Connecticut, USA

    Sarah Glaz

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Chapman, S.T., Freeze, M., Smith, W.W. (2000). On Generalized Lengths of Factorizations in Dedekind and Krull Domains. In: Chapman, S.T., Glaz, S. (eds) Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol 520. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3180-4_6

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  • DOI: https://doi.org/10.1007/978-1-4757-3180-4_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4835-9

  • Online ISBN: 978-1-4757-3180-4

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