Descending chains of modules and Jordan-Hölder theorem. (Q1882654)

The paper studies the relation between unique factorization domains, the Krull-Schmidt theorem and the Jordan-Hölder theorem, and how these conditions can be expressed in terms of freeness of commutative monoids. For instance, an integral domain \(R\) is a unique factorization domain iff \(R^*/U(R)\) is a free commutative monoid, where \(R^*\) is the multiplicative monoid of \(R\setminus\{0\}\) and \(U(R)\) is the group of invertible elements of \(R\). The authors aim to create a general framework for this kind of results. In this paper they consider the case of the Jordan-Hölder theorem, for which the approach is not fully successful, but nevertheless gives significant results. In the last section several examples are considered in the context of the present approach.