Modules, Monoids, and Factorizations

A key principle in studying mathematical objects is to understand their decompositions into the simplest possible parts. This principle is applicable to a wide spectrum of objects, ranging from elementary settings such as the decomposition of natural numbers into products of prime numbers--- questions that have been driving number theory since ancient times, over factorizations in more general rings (algebraic structures in which we can add and multiply), to the decomposition of modules into direct sums of indecomposable modules. These questions fall broadly under the purview of factorization theory, a subfield in the intersection of algebra, number theory, and additive combinatorics. The present project is concerned with the study of the interplay between factorization theory and module theory (the mathematical theory, very roughly, emerging as generalization of the study of solutions of linear equations over rings). This interplay emerges in two ways: (i) module theory provides a useful tool in the study of the factorization of elements in several important classes of noncommutative noetherian rings and algebras; in the present project in particular homological methods and Hopf algebras will be the focus of these investigations, and (ii) through the introduction of a suitable monoid, the study of direct-sum decompositions of modules can be approached with methods from factorization theory. In this second part, the focus will be on extending the existing successful monoid-theoretical approach, that is useful for finite direct sum decompositions, to infinite direct sum decompositions. We aim to understand the arithmetic of infinite direct sum decompositions and realization results as well as their limitations.