Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M_P over each field R_P (R localized at P), where P ranges over the minimal prime ideals of R. We assume that there exists a bound N_R on the ranks of all indecomposable finitely generated torsion-free R-modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is (r,r,…,r), then r∈{1,2,3,4,6}, even when R is not local.