The authors are motivated by the well-known theorem of Butler, saying that a torsion-free abelian group is a pure subgroup of a completely decomposable torsion-free group if and only if it is an epimorphic image of a completely decomposable group, and certain analogies between completely decomposable groups and direct sums of valuated torsion-free cyclic groups to prove (3.1 and 3.5): Let R be a principal ideal domain. An R-module G is a valuated submodule of a finite direct sum of cyclic valuated R-modules if and only if G is the epimorphic image of a finite direct sum of cyclic valuated modules. When \(R={\mathbb{Z}}\), the ring of integers, then a functor is constructed which assigns to every torsion- free valuated group H whose valuation satisfies obviously necessary restrictions, a torsion-free abelian group T(H) which contains H as a full-rank valuated subgroup. Using this device they show (4.3): Let G be a torsion-free group and H a finitely generated full subgroup of G. Then G is a pure subgroup (quotient) of a completely decomposable torsion-free group if and only if the valuated group H is a subgroup (quotient) of a direct sum of cyclic valuated groups. The theorem of Butler becomes a consequence of its analogue in valuated groups.
The paper contains more than this. An essential tool for the above results is a theorem of J. H. Moore [Lect. Notes Math. 874, 405-416 (1981; Zbl 0464.20041)] saying that every submodule of a finite direct sum of cyclic valuated R-modules is nice. The authors prove a theorem which implies Moore’s result. Moore also established a necessary condition which any valuated submodule of a direct sum of valuated cyclic R-modules must satisfy. This condition is generalized (5.1). The authors then characterize the discrete valuation rings R with the property that every rank-2 R-module satisfying Moore’s condition is in fact a valuated submodule of a direct sum of valuated cyclic modules. These are exactly the discrete valuation domains which are complete and have finite residue class fields. In the last section the connection between Warfield invariants and quasi-decomposability is studied for valuated modules with two generators over a discrete valuation domain.