Abstract :
Let R be a one-dimensional, reduced Noetherian ring with finite normalization, and suppose there exists a positive integer NR such that, for every indecomposable finitely generated torsion-free R-module M and every minimal prime ideal P of R, the dimension of MP, as a vector space over the localization RP (a field), is less than or equal to NR. For a finitely generated torsion-free R-module M, we call the set of all such vector-space dimensions the rank-set of M. What subsets of the integers arise as rank-sets of indecomposable finitely generated torsion-free R-modules? In this article, we give more information on rank-sets of indecomposable modules, to supplement previous work concerning this question. In particular we provide examples having as rank-sets those intervals of consecutive integers that are not ruled out by an earlier article of Arnavut, Luckas and Wiegand. We also show that certain non-consecutive rank-sets never arise.
