Almost Free Splitters

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds. For simplicity we will call such modules splitters, see [16]. Also other names like stones are used, see a dictionary in Ringel’s paper [14]. Our investigation continues [10]. In [10] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [10] we concentrated on splitters which are larger then the continuum and such that countable submodules are not necessarily free. The ‘opposite’ case of א1-free splitters of cardinality less or equal to א1 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 [12]. Contrary to the results in [10] and in accordance to [12] we can show that all א1-free splitters of cardinality א1 are free indeed.