Non-unique Factorizations, Ideal Theory, and Additive Theory

Non-Unique Factorizations. Let R be a noetherian domain. Then every nonzero element of R that is not a unit has a factorization into atoms (irreducible elements) of R. In general, there are many such decompositions, which differ not only up to units and the ordering of the factors. The main objective is to describe and classify the various phenomena of non-unique factorizations by arithmetical invariants and to relate these to the algebraic invariants of R. If a = u1 ... uk is such a factorization of an element into atoms, then k is called the length of the factorization, and the set L(a) of all possible factorization lengths is the set of lengths of a. Sets of lengths are finite and nonempty, and they are central arithmetical invariants. Multiplicative Ideal Theory. Its subject is the description of the multiplicative structure of an integral domain by means of ideals or certain systems of ideals (in technical terms, star and semistar operations, ideal and module systems). A main theme is the factorization of ideals (or of divisorial ideals, invertible ideals, and others) into prime ideals (or into radical ideals, primary ideals, and others). Additive (group and number) Theory. The principal objects of study are the sumsets of (mainly finite and nonempty) subsets A and B of an additive abelian group. The most common reoccurring theme is the idea that if the sumset is small, then A, B and their sumset must have structure. A further main object of study are subsequence sums and zero-sums of sequences over abelian groups, where a sequence (in this context) is a finite, unordered sequence allowing the repetition of elements. Problems dealing with sequences are often translated into problems with sets, and then they are studied via sumsets. The set of zero-sum sequences over a group (with concatenation as operation) is a Krull monoid. This Project is in the overlap of the above areas, and it is inspired by recent developments in them. Suppose that R is integrally closed. Then R is a Krull domain. Factoring elements is the same as factoring principal ideals, which lie inside the monoid of divisorial ideals. This gives rise to a transfer homomorphism from the elements of R to the monoid of zero-sum sequences over the class group of R, which preserves sets of lengths. Via this machinery we study sets of lengths in R with methods from Additive Theory. Furthermore, we study the ideal theory and sets of lengths in non-integrally closed domains.

Factorization Theory. In an atomic monoid (e.g., in a noetherian domain) every element has a factorization into atoms (irreducible elements). In general, there are many such (essentially distinct) decompositions. The main objective is to describe and classify the various phenomena of non-unique factorizations by arithmetical invariants and to relate these to the algebraic invariants of the underlying object. If a = u1 ... uk is such a factorization of an element into atoms, then k is called the length of the factorization, and the set L(a) of all possible factorization lengths is the set of lengths of a. Sets of lengths are central arithmetical invariants.Multiplicative Ideal Theory. Its subject is the description of the multiplicative structure of an integral domain (of a monoid) by means of ideals or certain systems of ideals (in technical terms, star and semistar operations, ideal and module systems). A main theme is the factorization of ideals (or of divisorial ideals, invertible ideals, and others) into prime ideals (or into radical ideals, primary ideals, and others).Additive (group and number) Theory. The principal objects of study are the sumsets (of mainly finite and nonempty subsets A and B of an additive abelian group), and subsequence sums, and zero-sums of sequences over abelian groups, where a sequence (in this context) is a finite, unordered sequence allowing the repetition of elements. The set of zero-sum sequences over a group (with concatenation as operation) is a Krull monoid.This Project was in the overlap of the above areas and gained from recent developments in them. We established new results on arithmetical invariants (in particular on sets of lengths). For the first time this was done also for non-commutative rings and for rings with zero-divisors.