Whitehead Modules over Domains

Let R be a commutative domain with 1. By a Whitehead module is meant an Rmodule M satisfying Ext₁ ^r(M, R) = 0. If R is such that.RD-submodules of torsion-free Whitehead modules are again Whitehead, then the hypothesis V = L makes it possible to reduce the problem of characterizing torsion-free Whitehead modules to Whitehead modules of cardinality ≤ ∣R∣.Proper Forcing is used to show that this criterion fails in ZFC.Applications are given to P.I.D.s of cardinality N₁countable valuation domains and almost maximal valuation domains of cardinality N₁.