Start:
April 1,2020
Alfred Geroldinger
FWF project number:
P33499-N

# Participants

Alfred Geroldinger, Ao.Univ.-Prof. Mag. Dipl.-Ing. Dr.
Professor
E-Mail:
alfred.geroldinger@uni-graz.at
Phone:
+43 (0)316 380 - 5154
Office:
Heinrichstraße 36, Room 526 (4th floor)
Homepage:
https://imsc.uni-graz.at/geroldinger
Andreas Reinhart, Priv.-Doz. Mag. Dr. rer. nat.
Postdoc
E-Mail:
andreas.reinhart@uni-graz.at
Phone:
+43 (0)316 380 - 5155
Office:
Heinrichstraße 36, Room 528 (4th floor)
Homepage:
http://imsc.uni-graz.at/reinhart
Qinghai Zhong, Dr.
Postdoc
E-Mail:
qinghai.zhong@uni-graz.at
Phone:
+43 (0)316 380 - 5155
Office:
Heinrichstraße 36, Room 528 (4th floor)
Homepage:
https://imsc.uni-graz.at/zhong

All participants are located at the

# 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. On product-one sequences over dihedral groups.
Journal of Algebra and its Applications, 21 (2022), 2250064.

2. On strongly primary monoids with a focus on Puiseux monoids.
J. Algebra, 567 (2021), 310-345.

3. On half-factoriality of transfer Krull monoids.
Communications in Algebra, 49 (2021), 409-420.

4. On an inverse problem of Erdős, Kleitman, and Lemke.
J. Combin. Theory Ser. A, 177 (2021), 105323.

5. On the incomparability of systems of sets of lengths.
European Journal of Combinatorics, to appear.

6. On a zero-sum problem arising from factorization theory.
In Combinatorial and Additive Number Theory IV, 11-24, Springer 2021.

7. On Clean, Weakly Clean, and Feebly Clean Commutative Group Rings.
Journal of Algebra and its Applications, 21 (2022), 2250085.

8. A realization result for systems of sets of lengths.
Israel Journal of Mathematics, 247 (2022), 177-193.

9. On product-one sequences over subsets of groups.
Period. Math. Hung., published online.

10. A characterization of length-factorial Krull monoids.
New York Journal of Mathematics, 27 (2021), 1347-1374.

11. On product-one sequences with congruence conditions over non-abelian groups.
Journal of Number Theory, 238 (2022), 253-268

12. On transfer homomorphisms of Krull monoids.
Bollettino dell'Unione Matematica Italiana, 14 (2021), 629-646

13. On the arithmetic of monoids of ideals.
Arkiv for Matematik, 60 (2022), 67-106

14. On transfer Krull monoids.
Semigroup Forum, 105 (2022), 73-95

15. 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.

16. On algebraic properties of power monoids of numerical monoids.
Israel Journal of Mathematics, to appear.

17. On Dedekind domains whose class groups are direct sums of cyclic groups.
submitted.

18. Valuation ideal factorization domains.
submitted.