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

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Rings, Polynomials, and Modules

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.

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Acknowledgements

This work was supported by the Austrian Science Fund FWF, Project Number P28864-N35, and the ANR project Caesar, project number ANR-12-BS01-0011

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Authors and Affiliations

  1. Institute for Mathematics, University of Graz, Heinrichstraße 36, 8010, Graz, Austria

    Alfred Geroldinger & Qinghai Zhong

  2. Laboratoire Analyse, Géométrie et Applications (LAGA, UMR 7539, CNRS), COMUE Université Paris Lumières, Université Paris 8, CNRS, 93526, Saint-Denis cedex, France

    Wolfgang A. Schmid

  3. Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, Université Paris 8, F-93430, Villetaneuse, France

    Wolfgang A. Schmid

Authors
  1. Alfred Geroldinger
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  2. Wolfgang A. Schmid
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  3. Qinghai Zhong
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Correspondence to Qinghai Zhong .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  2. Department of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Sophie Frisch

  3. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Sarah Glaz

  4. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Francesca Tartarone

  5. Dipartimento di Matematica, Università di Padova, Padova, Italy

    Paolo Zanardo

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Geroldinger, A., Schmid, W.A., Zhong, Q. (2017). Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids. In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_11

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