 Start:
 April 1,2020
 Project leader:
 Alfred Geroldinger
 FWF project number:
 P33499N
Participants
Professor
 EMail:
 alfred.geroldinger@unigraz.at
 Phone:
 +43 (0)316 380  5154
 Office:
 Heinrichstraße 36, Room 526 (4th floor)
 Homepage:
 https://imsc.unigraz.at/geroldinger
Postdoc
 EMail:
 andreas.reinhart@unigraz.at
 Phone:
 +43 (0)316 380  5155
 Office:
 Heinrichstraße 36, Room 528 (4th floor)
 Homepage:
 http://imsc.unigraz.at/reinhart
Postdoc
 EMail:
 qinghai.zhong@unigraz.at
 Phone:
 +43 (0)316 380  5155
 Office:
 Heinrichstraße 36, Room 528 (4th floor)
 Homepage:
 https://imsc.unigraz.at/zhong
All participants are located at the
University of Graz
Heinrichstraße 36
8010 Graz
Austria
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 noninvertible element has a factorization into irreducible elements, but in general there is no uniqueness. The main objective is to describe the various phenomena of nonuniqueness 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:
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 zerosum sequences over an abelian group. Monoids of zerosum 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

On productone sequences over dihedral groups.
Journal of Algebra and its Applications, 21 (2022), 2250064. 
On strongly primary monoids with a focus on Puiseux monoids.
J. Algebra, 567 (2021), 310345. 
On halffactoriality of transfer Krull monoids.
Communications in Algebra, 49 (2021), 409420. 
On an inverse problem of Erdős, Kleitman, and Lemke.
J. Combin. Theory Ser. A, 177 (2021), 105323. 
On the incomparability of systems of sets of lengths.
European Journal of Combinatorics, to appear. 

On a zerosum problem arising from factorization theory.
In Combinatorial and Additive Number Theory IV, 1124, Springer 2021. 
On Clean, Weakly Clean, and Feebly Clean Commutative Group Rings.
Journal of Algebra and its Applications, 21 (2022), 2250085. 
A realization result for systems of sets of lengths.
Israel Journal of Mathematics, 247 (2022), 177193. 
On productone sequences over subsets of groups.
Period. Math. Hung., published online. 
A characterization of lengthfactorial Krull monoids.
New York Journal of Mathematics, 27 (2021), 13471374. 
On productone sequences with congruence conditions over nonabelian groups.
Journal of Number Theory, 238 (2022), 253268 
On transfer homomorphisms of Krull monoids.
Bollettino dell'Unione Matematica Italiana, 14 (2021), 629646 
On the arithmetic of monoids of ideals.
Arkiv for Matematik, 60 (2022), 67106 
On transfer Krull monoids.
Semigroup Forum, 105 (2022), 7395 
On monoids of weighted zerosum sequences and applications to norm monoids in Galois number fields and binary quadratic forms.
Acta Math. Hungar., 168 (2022), 144185. 
On algebraic properties of power monoids of numerical monoids.
Israel Journal of Mathematics, to appear. 
On Dedekind domains whose class groups are direct sums of cyclic groups.
submitted. 
Valuation ideal factorization domains.
submitted.