Chains of free modules and construction of \(\aleph\)-free modules (Q1093685)

Let R be a commutative valuation domain (with unit) that is not a field. The class \({\mathcal F}_ 0\) consists of all countably generated torsion free R-modules. For \(n\geq 1\), \({\mathcal F}_ n\) consists of all modules M such that \(M=\cup_{\alpha <\omega_ n}F_{\alpha}\quad and\), for every limit ordinal \(\alpha\), \(F_{\alpha}=\cup_{i<\alpha}F_ i,\) where \(F_ 0=0\), each \(F_ i\) is a free module of rank at most \(\aleph_{n-1}\) that is a pure submodule of \(F_{i+1}\), and for every \(\alpha <\beta <\omega_ n\), \(F_{\beta}/F_{\alpha}\in {\mathcal F}_{n- 1}\). The author proves a number of results about such modules including the facts that every module in \({\mathcal F}_ n\) is \(\aleph_ n\)-free, and that there exists for every n an \(\aleph_ n\)-free module that is not \(\aleph_{n+1}\)-free. [For an infinite cardinal \(\beta\), an R-module M is \(\beta\)-free if every pure submodule K of rank \(<\beta\) can be embedded in a pure free submodule of M.]