Arithmetic of Rings and of their Ideals and Modules

In order to get a better understanding of mathematical objects one oftentimes factorizes (decomposes) objects into simpler parts which do not allow any further factorizations (decom- positions). We give two examples. Every positive integer can be written (in a unique way) as a product of primes (irreducible integers). Similarly, every polynomial with integer coefficients can be written (in a unique way) as a product of irreducible polynomials with integer coefficients. A ring is a central algebraic structure in mathematics. It is a set of elements with addition and multiplication which satisfy similar calculation rules as the set (the ring) of integers or the set (the ring) of all polynomials with integer coefficients. In an overwhelming number of interesting cases, every element of an abstract ring (or of a semigroup) also allows a factorization into irreducible elements. But in general the uniqueness property gets lost. To give an example, note that polynomials with non-negative integer coefficients can be factorized into irreducible polynomials with non-negative coefficients but such factorizations are not unique in general. The main goal of the present project is to study the non-uniqueness of factorizations, to describe it by arithmetical invariants, and to understand the arising phenomena from a structural point of view (this means to understand the interdependence of arithmetical invariants and the classical algebraic invariants of the underlying structures). Sets of lengths are key arithmetical invariants describing the non-uniqueness of factorizations. Let R be a ring and a an element of R. If a = u1 . . . uk , where k is a positive integer and u1 , . . . , uk are irreducible elements of R, then k is called a factorization length of a and the set L(a) of all possible factorization lengths is called the set of lengths of a. If there is one element b R for which L(b) contains more than one element, then for every positive integer n there is an element bn in the ring such that L(bn ) has more than n elements. Thus sets of lengths can become arbitrarily large. In many cases sets of lengths are a sort of generalized arithmetical progressions (an arithmetical progression is a set of the form {a, a + d, a + 2d, a + 3d, . . . , a + kd}, where d is the difference of the arithmetical progression). It is one of the goals of the present project to study the structure of sets of lengths for a large class of interesting rings. 1

This project was devoted to basic research in algebra (a subfield of mathematics). We first discuss its topics in everyday language and then we briefly discuss its results. In order to get a better understanding of mathematical objects one oftentimes factorizes (decomposes) objects into simpler parts which do not allow any further factorizations (decom- positions). We give two examples. Every positive integer can be written (in a unique way) as a product of primes (irreducible integers). Similarly, every polynomial with integer coefficients can be written (in a unique way) as a product of irreducible polynomials with integer coefficients. A ring is a central algebraic structure in mathematics. It is a set of elements with addition and multiplication which satisfy similar calculation rules as the set (the ring) of integers or the set (the ring) of all polynomials with integer coefficients. In an overwhelming number of interesting cases, every element of an abstract ring (or of a semigroup) also allows a factorization into irreducible elements. But in general the uniqueness property gets lost. To give an example, note that polynomials with non-negative integer coefficients can be factorized into irreducible polynomials with non-negative coefficients but such factorizations are not unique in general. Now let's get to the results. We studied the non-uniqueness of factorizations in a broad class of rings (more precisely and to give an example, in ideal semigroups of polynomial rings); described them by arithmetical invariants (more precisely, for example with the help of length sets); and thus we arrived at a better understanding of (some of) the arising phenomena from a structural point of view (more precisely, among others by progress in the Characterization Problem of Krull rings). Furthermore, we (in cooperation with a colleague from the US) started to work on a monograph on these topics.