### Project 03: "Additive theory, zero-sum theory and non-unique factorizations"

*This project is running since 2010.*

Principal investigator: Alfred Geroldinger

University of Graz, Austria.

*Mentor for*: McMahon; Barroero, Kreso, Weitzer.

### DK Student

**Second phase of the doctoral program:**-
**JunSeok Oh**(Republic of Korea; since October 2015)

*Email*: junseok.oh@uni-graz.at

*Mentors*: Karin Baur, Andreas Reinhart.

**First phase of the doctoral program:**-
**Daniel Smertnig**(Austria; March 2011–May 2014)

Personal homepage;*Email*: daniel.smertnig@uni-graz.at

*Mentors*: Christian Elsholtz, Robert Tichy.

*PhD Defense*: May 22, 2014.

*Referees*: A. Facchini (Padova), A. Geroldinger, A. Loper (Ohio).

*Examiners*: A. Facchini (Padova), A. Geroldinger

### Project description (pdf-file)

My research interests are in commutative
and non-commutative Algebra (with the focus on Multiplicative Ideal Theory of Prüfer,
Krull, and Mori rings and monoids [Gilmer 1992; Halter-Koch 1998]) and in Additive (Group
and Number) Theory (with the focus on Zero-Sum Theory and Addition Theorems; [Nathanson
1996; Grynkiewicz 2013]). Recurrent themes in my research are the arithmetic of rings and
semigroups (mainly semigroups of ideals and modules) and problems of non-unique factorizations
[Anderson 1997; Geroldinger and Halter-Koch 2006; Fontana et al. 2013; Chapman
et al. 2016]. Factorization theory is in the overlap of the above two areas [Geroldinger 2009].
Indeed, a most successful strategy is to use ideal theory of the monoid or domain to construct
transfer homomorphisms which shift the original algebraic problems to discrete combinatorial
problems. We give two examples of this strategy.

It is a classical result that, if one starts with Krull or Dedekind domains, arithmetic invariants
of such rings can be translated into zero-sum invariants over the class group. Deep
recent results (see [Smertnig 2013; Baeth and Smertnig 2015; Smertnig 2018]; and [Geroldinger
2016] for a survey), showed that the same is true for large classes of non-commutative
Dedekind domains (in particular, for maximal orders in central simple algebras over number
fields). Thus arithmetical invariants (including sets of lengths) of these non-commutative
Dedekind domains can be studied in the (commutative!) monoid of zero-sum sequences over
an associated abelian group.

Progress in module theory (pushed forward by Facchini and Wiegand; [Baeth and Wiegand
2013; Facchini 2006]) allows to study direct-sum decompositions of modules (in the non-Krull-
Schmidt case) with methods from Factorization Theory [Baeth and Geroldinger 2014].