A generalization of Whitehead’s problem and its independence

For certain classes of Dedekind domains S we want to characterize S-modules U such that Ext(U,M)=0 for some module S⊆M⊆Q. We shall call these modules M-Whitehead modules. On the one hand we will show that assuming (V=L) all M-Whitehead modules U are S0-free, i.e. U⊗S0 is a free S0-module where S0 is the nucleus of M. On the other hand if there is a ladder system on a stationary subset of ω1 that satisfies 2-uniformization, then there exists a non-S0-free M-Whitehead module. Conversely, we will show that in the special case of Abelian groups the existence of a non-S0-free R-Whitehead group–here R is a rational group–implies that there is a ladder system on a stationary subset of ω1 that satisfies 2-uniformization.