Multiplicative invariants and length functions over valuation domains
\(R\) denotes a commutative ring, \(\mathrm{Fin} R\) the class of finitely generated \(R\)-modules, and \((\Gamma,\cdot,\leq)\), denotes a commutative partially ordered semigroup. A map \(\mu: \mathrm{Fin} R\to\Gamma\) is called an invariant of \(\mathrm{Fin} R\), if, for isomorphic \(R\)-modules \(X, Y\), one has \(\mu(X)=\mu(Y)\). \(\mu\) is called a multiplicative invariant, if, for every short exact sequence \(0\to Y\to X\to X/Y\) in \(\mathrm{Fin} R\), the following conditions are satisfied: (i) \(\mu(X)=\mu(Y)\mu(X/Y)\); (ii) \(\mu(Y)\leq \mu(X)\); (iii) if \(Z\) is a homomorphic image of \(X\), then \(\mu(Z)\leq \mu(X)\). For a valuation domain \(R\), \(\mathcal I=\mathcal I(R)\) denotes the multiplicative semigroup of the ideals of \(R\), linearly ordered by \(I<J\) iff \(I\supset J\); then a map \(\mu_\mathcal I:\mathrm{Fin} R\to\mathcal I(R)\) is defined as follows: given a finitely generated \(R\) module \(X\), with some ordered basis \(\mathcal G\), then \(\mu_\mathcal I(X)=B_1\cdots B_n\), where \(B_1,\dots, B_n\) is the \(\mathcal G\)-annihilator set of \(X\).NEWLINENEWLINESome of the results are as follows:NEWLINENEWLINETheorems 2.4-5. Let \(R\) be a valuation domain. Then the map \(\mu_\mathcal I\) is a well-defined multiplicative invariant of Fin\(R\). Moreover, this map has the universal property in the class of valuative invariants. The author's starting point was the notion of length function of Mod\(R\), introduced by \textit{D. G. Northcott} and \textit{M. Reufel} [Q. J. Math., Oxf. II. Ser. 16, 297--321 (1965; Zbl 0129.02203)] that was meant to be a generalization of the classical Jordan-Hölder length of modules. The present author's interest is also in various length functions. One of the products of this paper is an alternative description of length functions (obtained by Northcott and Reufel), namely via use of the map \(\mu_\mathcal I\),
scientific article; zbMATH DE number 6030534