Doctoral School Mathematics and Scientific Computing

Abstract: In a noetherian integral domain, every nonzero non-unit can be written as a finite product of irreducible elements. Such a representation is unique if the ring is factorial, but there is a huge class of interesting rings where such factorizations exist but are not unique. It is a goal of factorization theory to measure this non-uniqueness. For example, to a fixed element a of the ring one may associate its set of lengths, consisting of all the lengths of factorizations of a into irreducibles. One can then study the structure of these sets for a given ring.

Rings where such problems have been considered include in particular the maximal orders in number fields, for example the ring Z[\exp(2\pi i/n)] as a subring of the cyclotomic field Q(\exp(2\pi i/n). We consider the non-commutative analogue of such rings, namely maximal orders in central simple algebras over number fields. This class includes, for example, maximal orders in matrix rings and in quaternion algebras (over number fields). [hide abstract]