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Project summary

The present project lies in the overlap of factorization theory (considered as a subfield of ring and module theory) and additive combinatorics. The term Combinatorial Factorization Theory refers to factorization theory of Krull monoids (and of transfer Krull monoids as their generalization), and to all associated aspects of additive combinatorics.

A commutative integral domain (more generally, a commutative cancellative monoid) is factorial if and only if it is Krull with trivial class group. The main objective is to describe the non-factoriality of Krull monoids with nontrivial class group by arithmetic invariants, such as length sets.

It is well-known that there is a transfer homomorphism from a Krull monoid to the associated monoid of zero-sum sequences over its class group. Monoids of zero-sum sequences are studied with methods from additive combinatorics (and this is what we are going to do) and arithmetic invariants (such as length sets) can be pulled back from the monoid of zero-sum sequences to the original Krull monoid.

We study two innovative questions. (i) As mentioned above, it is well-known that length sets depend on the class group only. We study the associated inverse question. Suppose that all length sets of two Krull monoids with finite class groups $G$ and $G'$ coincide. Does this imply that $G$ and $G'$ are isomorphic? (ii) There are rings and monoids, that are not Krull, but allow a transfer homomorphism to a monoid of zero-sum sequences. Rings and monoids with this property are called transfer Krull. They need neither be commutative nor cancellative nor $v$-noetherian (indeed, examples include non-commutative HNP rings and non-cancellative monoids of modules). The reason why or why not certain monoids are transfer Krull is not understood at all. We will carry out a first systematic study of this phenomenon.