Factorizations of algebraic integers

By the Fundamtental Theorem of Arithmetic, every positive integer is a product of primes in an essentially unique way. In the ring of integers of an algebraic number field, also every non-zero non-unit is a product of irreducible elements (atoms), but in general there are many essentially distinct such representations. In classical algebraic number theory of the 19th century, this failure of unique factorization led to Dedekind`s ideal theory and to Kronecker`s divisor theory. Only in the late 20th century, inspired by Carlitz` characterization of algebraic number fields of class number 2, W. Narkiewicz began a systematical combinatorial and analytical investigation of phenomena of non-unique factorizations in rings of integers of algebraic number fields. In the sequel, it turned out the same kind of questions are not only of interest for algebraic integers, but equally for more general integral domains, for (cancellative) monoids, for zero-sum sequences over abelian groups and for several categories of finitely generated modules. This led to a general theory of non-unique factorizations. This theory is one of the focus areas of the working group of Algebra and Number Theory of the University of Graz, and also the current project is devoted to this theory of non-unique factorizations. The following problems are in the center of interest fort he project under application: Algebraic integers with well-structured sets of factorizations and their distribution and frequency in orders of algebraic number fields; the structure of half-factorial orders in algebraic number and algebraic function fields; the connection between phenomena of non-unique factorizations (in particular, of sets of lengths) and the structure of the class group; the connection between non-unique factorizations and the Kronecker and Nagata rings (for an arbitrary integral domain). Phenomena of non-unique factorizations will be studied in appropriate auxiliary monoids (finitely generated and finitely primary monoids, C-monoids), and then the results will be applied using transfer principles. The sructure of zero-sum sequences in finite abelian groups. In particular, we want to investigate half-factorial subsets of finite abelian groups and the combinatorial invariants defined by them; the structure of zero-sum sequences with extemal properties (in particular for abelian groups of rank 2); the combinatorial invariants of Additive Group Theory which occuring in the asymptotic formulas of the analytic theory on non-unique factorizations of algebraic integers.

By the Fundamtental Theorem of Arithmetic, every positive integer is a product of primes in an essentially unique way. In the ring of integers of an algebraic number field, also every non-zero non-unit is a product of irreducible elements (atoms), but in general there are many es-sentially distinct such representations. In classical algebraic number theory of the 19th centu-ry, this failure of unique factorization led to Dedekind`s ideal theory and to Kronecker`s divi-sor theory. Only in the late 20th century, inspired by Carlitz` characterization of algebraic number fields of class number 2, W. Narkiewicz began a systematical combinatorial and analytical investi-gation of phenomena of non-unique factorizations in rings of integers of algebraic number fields. In the sequel, it turned out the same kind of questions are not only of interest for alge-braic integers, but equally for more general integral domains, for (cancellative) monoids, for zero-sum sequences over abelian groups and for several categories of finitely generated mod-ules. This led to a general theory of non-unique factorizations. This theory is one of the focus areas of the working group of Algebra and Number Theory of the University of Graz, and also the current project is devoted to this theory of non-unique factorizations. The following problems are in the center of interest fort he project under application: Algebraic integers with well-structured sets of factorizations and their distribution and fre-quency in orders of algebraic number fields; the structure of half-factorial orders in algebraic number and algebraic function fields; the connection between phenomena of non-unique fac-torizations (in particular, of sets of lengths) and the structure of the class group; the connec-tion between non-unique factorizations and the Kronecker and Nagata rings (for an arbitrary integral domain). Phenomena of non-unique factorizations will be studied in appropriate auxiliary monoids (finitely generated and finitely primary monoids, C-monoids), and then the results will be applied using transfer principles. The sructure of zero-sum sequences in finite abelian groups. In particular, we want to inves-tigate half-factorial subsets of finite abelian groups and the combinatorial invariants defined by them; the structure of zero-sum sequences with extemal properties (in particular for abelian groups of rank 2); the combinatorial invariants of Additive Group Theory which occuring in the asymptotic formulas of the analytic theory on non-unique factorizations of algebraic inte-gers. Direct-sum decompositons of finitely generated modules. The deviation from the Theorem of Krull-Remak- Schmidt is interpeted as an phenomenon of non-unique factorizations and thus becomes a subject to be studied with the general theory of non-unique factorizations. We will investigate cancellation from direct sums (both torsion-free and mixed cancellation), in parti-cular for finitely generated modules over orders in algebraic number fields; construction prin-ciples for (big) indecomposable modules (under as general conditions as possible); the struc-ture of categories of modules which lead to Krull monoids and the connection between the arithmetic of the Krull monoid and the ideal-theoretical properties of the base ring.