Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings RR. We assume there exists a positive integer NRNR such that, for every indecomposable RR-module MM and for every minimal prime ideal PP of RR, the dimension of MPMP, as a vector space over the field RPRP, is less than or equal to NRNR. If a nonzero indecomposable RR-module MM is such that all the localizations MPMP as vector spaces over the fields RPRP have the same   dimension rr, for every minimal prime PP of RR, then r=1,2,3,4r=1,2,3,4 or 6. Let nn be an integer ≥8≥8. We show that if MM is an RR-module such that the vector space dimensions of the MPMP are between nn and 2n−82n−8, then MM decomposes non-trivially. For each n≥8n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from nn to 2n−72n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.