ON TORSION-FREE MODULES OVER VALUATION DOMAINS

After transferring the concept of a basic subgroup from abelian group theory to torsion-free modules over valuation domains, we investigate three classes of indecomposable torsion-free modules over almost maximal valuation domains. The most attractive class is the one of purely indecomposable modules, i.e., modules all of whose pure submodules are indecomposable. We obtain a complete and independent set of invariants for this class. If n is the basic rank of some torsion-free module, the reduced part of its n-th exterior power is purely indecomposable. We give a necessary and sufficient criterion involving this exterior power for the module in question to be totally indecomposable. A module is totally indecomposable if every pure submodule is either completely decomposable or indecomposable. Co-purely indecomposable modules are also discussed. The basic rank of these modules is one less than their finite rank. We investigate the endomorphism rings of the members of all three classes. In particular, a counter-example to a theorem of D. M. Arnold {2} is given.;Finally we turn our attention to the question when the module R('R) is separable. Under additional assumptions we can show that this is the case if and only if the valuation domain R is either discrete of maximal. In particular, this answers a question raised by B. Zimmermann in {19}.