Ideal factorization in commutative rings and monoids

It is important to study generalized domains of numbers, which are extensions of the integers (like the Gaussian integers) to answer various questions in number theory. For instance, it is possible to solve the problem of representing a positive integer as a sum of squares by using the Gaussian integers. The elements in these extensions are less handsome than the integers, since it may be impossible to decompose them into a product of generalized prime numbers. A way of solving this problem is to consider special sets of numbers, so called ideals. It is possible to add and multiply (classical) ideals, like numbers. Therefore, the question arises whether it is possible to decompose ideals into products of other ideals. It is of particular interest to study decompositions of ideals into prime ideals and their generalizations, since they are some sort of generalized prime numbers. The goal of the research project is to study product decompositions of ideals (also called ideal factorizations) into generalizations of prime ideals and to use this knowledge to study product decompositions into elements. Many scientific papers have dealt with this problem. They considered, for instance, decompositions into primary ideals and radical ideals. This research will put special emphasis on proving results with vast generality. It is possible to consider other ideal theories which differ from the classical ideal theory (e.g. the v-ideal theory). These ideal theories tend to be more suitable to describe element decompositions. There are also several phenomena that cannot occur in the classical situation. It is possible to do ideal theory in a purely multiplicative way (i.e., without using an addition). Therefore, it is obvious to investigate objects (so called monoids) with a purely multiplicative structure. One method for solving these problems is the classification of the studied objects by using the existing literature. More precisely, one looks for properties which are satisfied by these (and only these) objects. Another method is to study the discrepancy between ideal factorizations and element factorizations by using so called class groups. Last but not least, it is important to provide a large number of examples to enclose the studied objects. The peculiarity in this research project is to investigate certain problems in a setting that has not been considered before. Special emphasis is put on gaining knowledge which is helpful for a better understanding of the classical situation. It will also be used to supplement and deepen the current knowledge about ideal factorizations.

Ideal factorization in commutative rings and monoids Summary for public relations The project was in the overlap of the areas of multiplicative ideal theory and factorization theory. First we give a brief overview on the (project relevant) terminology. It is well known that every positive integer can be written as a finite product of prime numbers (resp. prime elements). This fact is also known as the fundamental theorem of arithmetic. In general, integral domains whose elements possess the aforementioned type of product decompositions are called factorial domains. Besides the decomposition theory for elements of an integral domain, one can also develop a decomposition theory for certain sets of elements of integral domains. These sets of elements are called ideals. There is also an ideal theoretic analog of factorial domains, namely the so called Dedekind domains. More precisely, an integral domain is called a Dedekind domain if each of its ideals can be written as a product of prime ideals. The project's main achievement was the study of various generalizations of factorial domains and Dedekind domains. One possibility to generalize them is to replace prime elements resp. prime ideals by radical elements resp. radical ideals. Another possibility is the investigation of other types of structures (like cancellative monoids). For instance, the positive integers (together with multiplication) form a cancellative monoid. For cancellative monoids, there also exists an extensive ideal theory and different types of product decompositions (of their elements and ideals). There is a variety of known characterizations of factorial domains and Dedekind domains. For instance, an integral domain is a Dedekind domain if and only if each of its nonzero ideals is invertible. It was an important project goal to find similar characterizations of the abstracted notions. For example, we described SP-domains and w-SP-monoids (which are an extension of Dedekind domains). Furthermore, we characterized radical factorial monoids and valuation factorization domains. (The aforementioned concepts extend the class of factorial domains.) It was another project achievement to investigate special constructions (like monoid rings and Nagata rings) with respect to the aforementioned variations of factorial domains and Dedekind domains. As a consequence of these investigations, we were able to provide a substantial class of nontrivial examples of w-SP-domains and radical factorial domains. Besides providing the existence of certain factorizations (i.e., product decompositions) and characterizing them, it was also a project goal to describe the possible structure of these decompositions (into elements and ideals). We introduced and studied a variety of different invariants (like the catenary degree or the elasticity) which describe the properties of factorizations. As part of the project, we were able to determine a number of these invariants (of monoids of ideals) of orders in quadratic number fields.