Torsion-free modules of finite rank over a discrete valuation ring

Finite direct sums of finite rank purely indecomposable modules (called pi-decomposable modules) are characterized by the ranks of quotients of radicals. Torsion-free homomorphic images of pi-decomposable modules form an interesting class admitting a nice homological characterisation. Torsion-free modules quasi-isomorphic to a pi-decomposable module have properties reminiscent of almost completely decomposable abelian groups, including characterizations of representation type via correspondences with representations of finite partially ordered sets. This leads to the construction of indecomposable modules.