On modules over valuation domains whose finitely generated submodules are direct sums of cyclics
Let \({\mathcal C}\) denote the class of all \(R\)-modules over a valuation domain \(R\), all finitely generated submodules of which are direct sums of cyclic \(R\)-modules. The authors prove that a torsion module \(M\) lies in \({\mathcal C}\) (in general, a module M belongs to \({\mathcal C}\) iff its torsion part belongs to \({\mathcal C})\) iff every countably generated submodule of \(M\) is isomorphic to a submodule of a direct sum of uniserial \(R\)-modules, iff \(M\) is a submodule of a pseudo-separable \(R\)-module (\(K\) is pseudo-separable if every finite subset of \(K\) is embeddable in a cyclically pure submodule that is a finite direct sum of uniserial submodules).
scientific article; zbMATH DE number 4049753
scientific article