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Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids

In: Rings, Polynomials, and Modules

Author

Listed:
  • Alfred Geroldinger

    (University of Graz, Institute for Mathematics)

  • Wolfgang A. Schmid

    (COMUE Université Paris Lumières, Laboratoire Analyse, Géométrie et Applications (LAGA, UMR 7539, CNRS)
    Sorbonne Paris Cité, Université Paris 13)

  • Qinghai Zhong

    (University of Graz, Institute for Mathematics)

Abstract

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

Suggested Citation

  • Alfred Geroldinger & Wolfgang A. Schmid & Qinghai Zhong, 2017. "Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids," Springer Books, in: Marco Fontana & Sophie Frisch & Sarah Glaz & Francesca Tartarone & Paolo Zanardo (ed.), Rings, Polynomials, and Modules, pages 191-235, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-65874-2_11
    DOI: 10.1007/978-3-319-65874-2_11
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