Non-unique Factorization and zero-sum sequences

By the well-known Fundamental Theorem of Arithmetic every positive integer has a unique factori-zation into a product of primes. This theorem is no longer true in rings of integers of algebraic number fields and in several other structures in which yet every element has a factorization in-to irreducibles but not in an essentially unique way. In the classical algebraic number theory of the 19th century it was shown how the intoduction of ideal objects my help to overcome this non-uniqueness. The most important concepts in this connection are Richard Dedekind`s theory of ideals, Leopold Kronecker`s theory of divisors and Kurt Hensel`s theory of p-adic numbers. Only in the second half of the 20th century, phenomena of non-unique factorizations were studied for their own sake. The starting point were the investigations of L. Carlitz and W. Narkiewicz in the theory of algebraic numbers and those of the brothers D.D. and D.F. Anderson in the theory of commutative rings. Since many years the theory of non-unique fac-torizations has been one of the main themes of the group of Algebra and Number Theory in the Institute of Mathematics and Scientific Computing of the Karl-Franzens-University of Graz. During the project under review we continued the investigations of non-unique factorizations in an extensive way. Besides the classical objects of algebraic number theory, the theory of zero-sum sequences was one of the main objectives. That theory is not only an essential tool fort he investigations of algebraic integers, it has also an independent tradition in additive number theory and combinatorics ("additive group theory"). In the course of this project the monograph "Non-Unique Factori-zations. Algebraic, Combinatori-al and Analytic Theory" (700pp.) by A. Geroldinger und F. Halter-Koch was completed. This volume is the first overall presentation of the various aspects and the foundations of the theory of non-unique factorizations. It comprises many results obained during this project and published here for the first time. The most important mathematical results of the project concern the size and structure of half-factorial subsets of finite abelian groups, the algebraic and arithmetical theory of C-monoids, the frequency of algebraic integers with a cleary arranged set of factorizations and the explicit description of the Krull monoid built from the category of finitely generated modules over certain local noetherian rings.