Exterior Powers and Torsion-Free Modules over Almost Maximal Valuation Domains
The study of torsion-free modules of finite rank over almost maximal valuation domains has been initiated by L. Fuchs and G. Viljoen in . 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 . 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 : 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 .
Unable to display preview. Download preview PDF.
- L. Fuchs, Infinite abelian groups, vol. 2, Academic Press (1973).Google Scholar
- L. Fuchs and L. Salce, Prebasic submodules over valuation rings, Ann. Mat. Pura Appl. (to appear).Google Scholar
- L. Fuchs and G. Viljoen, On finite rank torsion-free modules over almost maximal valuation domains, Comm. Alg. (to appear).Google Scholar
- M. Marcus, Finite dimensional multilinear algebra, vol. 2, Marcel Dekker, Inc. (1975).Google Scholar