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].KeywordsValuation RingEndomorphism RingFinite RankIndecomposable ModuleDiscrete Valuation RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.