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 subset of M subset of 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 S-0-free, i.e. U circle times S-0 is a free S-0-module where S-0 is the nucleus of M. On the other hand if there is a ladder system on a stationary subset of col that satisfies 2-uniformization, then there exists a non-S-0-free M-Whitehead module. Conversely, we will show that in the special case of Abelian groups the existence of a non-So-free R-Whitehead group-here R is a rational group-implies that there is a ladder system on a stationary subset of omega(1) that satisfies 2-uniformization. (C) 2007 Elsevier B.V. All rights reserved.