Exterior powers and torsion-free modules over almost maximal valuation domains

Abstract

The study of torsion-free modules of finite rank over almost maximal valuation domains has been initiated by L. Fuchs and G. Viljoen in [6]. They investigated purely indecomposable modules, i.e. modules all of whose pure submodules are indecomposable. This is perhaps the most tractible class of indecomposable modules: they can be classified in terms of certain invariants and their endomorphism rings are valuation domains, especially they are local. There is a way to assign to each torsion-free module M of finite rank a purely indecomposable module: if n is the basic rank of M, then the reduced part of the n-th exterior power of M, AnM, is purely indecomposable. The problem how to read off module-theoretic properties of M in AnM will be settled for the class of totally indecomposable modules. This class was introduced by D. M. Arnold in [2]. A module M is called totally indecomposable if it is not a direct sum of rank 1 modules and every pure submodule is either completely decomposable or indecomposable. We obtain the following result generalizing theorem 2.1 in [2]: M is totally indecomposable iff every nonzero decomposable element in AnM is not divisible by all ring elements and M is not completely decomposable. This characterization enables us to construct a totally indecomposable module M of rank k and of basic rank 2, whose endomorphism ring is not local. Thus we obtain a counter-example to theorem 1.7 in [2].