Non-unique factorizations in noncommutative rings

Already from school we know that every integer can be written in a unique way as a product of prime numbers (and a sign). The prime numbers are characterized in particular by the fact that they cannot themselves be written as products of smaller factors,that is, that they are atoms. In mathematics we encounter an abundance of other objects in which we can multiply and add according to the usual rules. In algebra, such an object is called a ring. In many of these rings, it is again possible to express each element as a product of atoms. Usually, however, such a representation is far away from being unique. In many cases it is even possible for an element to possess representations (factorizations) of different lengths. In factorization theory, one tries to study this behavior qualitatively and quantitatively. This subfield of algebra and number theory has its origins in the study of factorizations of algebraic numbers. Nowadays, one often looks at much more general rings. Most of the time these rings are commutative, that is, rings in which the order of the factors in a product is irrelevant. Only a few years ago, the study of these type of questions has been initiated in noncommutative rings, that is, rings in which a product does depend on the order of its factors. In this area, there remain many fundamental open questions, which will be investigated in the course of the present project, non-unique factorizations in noncommutative rings. For instance, one expects that in a well-behaved ring, each element has at most finitely many different factorization lengths. One goal of the current project is to give easily applicable sufficient conditions for this property, similar to the ones we already have for commutative rings. The elasticity of an element is the ratio of the longest factorization length to the shortest one. The elasticity provides a measure of the non-uniqueness of the factorizations of an element. A bigger elasticity means that the element is further removed from having a unique factorization. Another goal of the present project consists of determining how big the elasticity can become in certain important rings that are of classical interest in noncommutative ring theory. This includes in particular quaternion orders, certain rings of differential operators (Weyl algebras) and quantized coordinate rings of the plane respectively the torus.

In this project the connection between arithmetical and algebraic properties in, predominantly noncommutative, rings was studied. The project has been carried out in international collaboration with colleagues at Dartmouth College (United States) and the University of Waterloo (Canada). A first type of questions concerns the factorization theory in noncommutative. Already from school we know that every integer can be written in a unique way as a product of prime numbers (and a sign). The prime numbers are characterized in particular by the fact that they cannot themselves be written as products of smaller factors,that is, that they are atoms. In mathematics we encounter an abundance of other objects in which we can multiply and add according to the usual rules. In algebra, such an object is called a ring. In many of these rings, it is again possible to express each element as a product of atoms. Usually, however, such a representation is far away from being unique. In many cases it is even possible for an element to possess representations (factorizations) of different lengths. In the project we classified the Hermite property of quaternion orders, a class of noncommutative rings. This algebraic (module-theoretic) property yields a dichotomy between two completely different factorization theory regimes: in one case the arithmetic of the quaternion orders is very tame in that arithmetical invariants are all finite and classic techniques apply; in the other case the arithmetical invariants are infinite. In addition, the factorization theory of acyclic cluster algebras has been fully determined. Cluster algebras are a class of rings that have been the focus of intense study over the past two decades, having connections to combinatorics, representation theory, and other branches of mathematics. A second focus was on (noncommutative) rational power series and the connection between arithmetic properties of their coefficient sequences and the structural decompositions of such series. We showed that rational power series (with coefficients in the rational numbers) have a particular simple structure if there are only finitely many prime numbers dividing their coefficients. This extends a result from the univariate setting, originally due to Plya, to multivariate power series and and confirms a 40 year old conjecture. Rational powers series also play a role in theoretical computer science, since they describe the behavior of weighted finite automatons and our result characterizes the unambiguous weighted finite automata, i.e., a class having a particular simple structure.