An indepedence result on cotorsion theories over valuation domains

It is shown that, over suitable valuation domains R with field of quotients Q, the cotorsion theory G(K) generated by K = Q/R coincides with the cotorsion theory Cr-partial derivative cogenerated by the Fuchs' divisible module partial derivative, provided that Godel's Axiom of Constructibility V = L is assumed. On the other hand, assuming Martin's Axiom and the negation of the Continuum Hypothesis, it is proved that the cotorsion theory G(K) is strictly smaller than C-partial derivative. by exhibiting a strongly (N - k )-free divisible module M of projective dimension 2 such that Ext(R)(1) (M, K) = 0. Applications to Whitehead R modules are derived