Bounds for indecomposable torsion-free modules

Let MM be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring RR with finitely generated normalization. The rank   of MM is the tuple of vector-space dimensions of MPMP over each field RPRP (RR localized at PP), where PP ranges over the minimal prime ideals of RR. We assume that there exists a bound NRNR on the ranks of all indecomposable finitely generated torsion-free RR-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 RR a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module MM is (r,r,…,r)(r,r,…,r), then r∈{1,2,3,4,6}r∈{1,2,3,4,6}, even when RR is not local.