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Project summary

The present project is devoted to the ideal and module theory and to the arithmetic of (mainly commutative) noetherian domains, Mori domains, and Krull domains. Its main theme is factorization theory, in other words the study of the arithmetic of rings, of their semigroups of (invertible, divisorial, and other) ideals, and of semigroups of modules over these rings.

Let $H$ be a commutative monoid satisfying a weak cancellation property and the ascending chain condition on principal ideals. Then every non-invertible element has a factorization into irreducible elements, but in general there is no uniqueness. The main objective is to describe the various phenomena of non-uniqueness by arithmetical invariants (such as sets of lengths, catenary and tame degrees) and to study the interdependence of arithmetical invariants and classical algebraic invariants of the underlying algebraic structures. The following monoids are in the center of interest:

Transfer Krull monoids: they include all commutative Krull domains but also wide classes of non-commutative Dedekind domains.

Mori domains $R$ with nonzero conductor with respect to their complete integral closure.

Semigroups of ideals of the domains mentioned above.

Additive monoids of isomorphism classes of certain types of modules under direct sums.

A key strategy runs as follows. Using ideal and module theory of the object $H$ one constructs a transfer homomorphism from $H$ to a simpler combinatorial object $B$. Transfer homomorphisms allow to pull back arithmetical information from $B$ to $H$. In case of a transfer Krull monoid $H$ the simpler object $B$ is a monoid of zero-sum sequences over an abelian group. Monoids of zero-sum sequences are studied with methods from additive combinatorics and, indeed, the project has a strong overlap with this area. The arithmetic of semigroups of ideals and modules, that are not cancellative, will be studied for the first time in a systematic way.

Publications

1. arxiv doi
   A. Geroldinger, D. Grynkiewicz, Jun Seok Oh, and Qinghai Zhong
   On product-one sequences over dihedral groups.
   Journal of Algebra and its Applications, 21 (2022), 2250064.
2. arxiv doi
   A. Geroldinger, F. Gotti and S. Tringali
   On strongly primary monoids with a focus on Puiseux monoids.
   J. Algebra, 567 (2021), 310-345.
3. arxiv doi
   Weidong Gao, Chao Liu, S. Tringali, and Qinghai Zhong
   On half-factoriality of transfer Krull monoids.
   Communications in Algebra, 49 (2021), 409-420.
4. arxiv doi
   Qinghai Zhong
   On an inverse problem of Erdős, Kleitman, and Lemke.
   J. Combin. Theory Ser. A, 177 (2021), 105323.
5. arxiv doi
   A. Geroldinger and W.A. Schmid
   On the incomparability of systems of sets of lengths.
   European Journal of Combinatorics, 111 (2023), 103694.
6. arxiv doi
   Aqsa Bashir, A. Geroldinger, and A. Reinhart
   On the arithmetic of stable domains.
   Communications in Algebra, 49 (2021), 4763-4787.
7. arxiv doi
   Aqsa Bashir, A. Geroldinger, and Qinghai Zhong
   On a zero-sum problem arising from factorization theory.
   In Combinatorial and Additive Number Theory IV, 11-24, Springer 2021.
8. arxiv doi
   Y. Li and Qinghai Zhong
   On Clean, Weakly Clean, and Feebly Clean Commutative Group Rings.
   Journal of Algebra and its Applications, 21 (2022), 2250085.
9. arxiv doi
   A. Geroldinger and Qinghai Zhong
   A realization result for systems of sets of lengths.
   Israel Journal of Mathematics, 247 (2022), 177-193.
10. arxiv doi
    Victor Fadinger and Qinghai Zhong
    On product-one sequences over subsets of groups.
    Period. Math. Hung., 86 (2023), 454–494.
11. arxiv link
    A. Geroldinger and Qinghai Zhong
    A characterization of length-factorial Krull monoids.
    New York Journal of Mathematics, 27 (2021), 1347-1374.
12. pdf doi
    Kevin Zhao and Qinghai Zhong
    On product-one sequences with congruence conditions over non-abelian groups.
    Journal of Number Theory, 238 (2022), 253-268
13. arxiv doi
    A. Geroldinger and F. Kainrath
    On transfer homomorphisms of Krull monoids.
    Bollettino dell'Unione Matematica Italiana, 14 (2021), 629-646
14. pdf doi
    A. Geroldinger and M. Azeem Khadam
    On the arithmetic of monoids of ideals.
    Arkiv for Matematik, 60 (2022), 67-106
15. arxiv doi
    Aqsa Bashir and Andreas Reinhart
    On transfer Krull monoids.
    Semigroup Forum, 105 (2022), 73-95
16. arxiv doi
    A. Geroldinger , F. Halter-Koch, and Q. Zhong
    On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms.
    Acta Math. Hungar., 168 (2022), 144-185.
17. pdf
    Pierre-Yves Bienvenu and A. Geroldinger
    On algebraic properties of power monoids of numerical monoids.
    Israel Journal of Mathematics, to appear.
18. pdf doi
    Gyu Whan Chang and A. Geroldinger
    On Dedekind domains whose class groups are direct sums of cyclic groups.
    J. Pure Appl. Algebra, 228 (2024), 107470.
19. arxiv doi
    Laura Cossu and Salvatore Tringali
    On the finiteness of certain factorization invariants.
    Arkiv för Matematik, 62 (2024), 21-38.

Last changed: 2023-1-1