Almost free splitters

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that ${\displaystyle Ex{t}_R(G,G)=0}$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel's paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger than the continuum and such that countable submodules are not necessarily free. The "opposite" case of ${\displaystyle {\aleph }_1}$-free splitters of cardinality less than or equal to ${\displaystyle {\aleph }_1}$ 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 Hausen [7]. In contrast to the results of [5] and in accordance with [7] we can show that all ${\displaystyle {\aleph }_1}$-free splitters of cardinality ${\displaystyle {\aleph }_1}$ are free indeed.