Investigations on the behaviour of torsion-free reduced modules over complete discrete valuation rings have been previously carried out by these and other authors in ZFC set theory. In this paper similar investigations are made, working under the additional set-theoretical hypothesis \(V=L\). The modules involved here are strongly \(\chi\)-free quâ \(A\)-modules, where \(A\) is an algebra. The usual step-lemmas associated with a construction in \(V=L\) are derived and used in the proof of theorem 1, which maintains that if \(R\) is a complete discrete valuation ring and \(A\) is a unital, torsion-free, Hausdorff (in the \(p\)- adic topology) \(R\)-algebra, then, for each regular, not-weakly compact cardinal \(\chi>| A|\), there exists a strongly \(\chi\)-free \(A\)- module \(H\) of cardinality \(\chi\) such that \(E_ R(H)=A\otimes\text{Ines} H\). A second theorem is proved and is generalized in theorem 3, of which an indication of proof is given.