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

# Participants

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

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

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 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:

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.