Almost free splitters

Abstract

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext_R(G,G)=0 holds. For simplicity we will call such modules splitters. Our investigation continues math.LO/9910159. In math.LO/9910159, we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In math.LO/9910159 we concentrated on splitters which are larger then the continuum and such that countable submodules are not necessarily free. The `opposite' case of aleph_1-free splitters of cardinality less or equal to aleph_1 was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by a result of Hausen. We can show that all aleph_1-free splitters of cardinality aleph_1 are free indeed.