Essentially indecomposable modules over a complete discrete valuation ring (Q5902977)

Torsion-free modules over a complete discrete valuation ring R are markedly different from abelian groups or modules over an incomplete discrete valuation ring in that the only indecomposable modules which exist have rank 1 and so are isomorphic to R itself or the field, Q, of fractions of R [\textit{I. Kaplansky}, ''Infinite abelian groups'' (1954; Zbl 0057.019); p. 45]. In this paper we investigate how close a reduced torsion-free R-module of infinite rank can come to being indecomposable. In particular we establish in {\S} 4 the existence of essentially indecomposable modules with basic submodules of countable rank. The results here bear a strong resemblance to results on p-groups.