几乎自由的分离器
Almost free splitters
摘要：
设R是有理数的子环。我们想研究自分裂R-模G满足Ext_R（G，G）=0.为了简单起见，我们将这些模块称为分离器。我们的调查还在继续数学。LO/9910159。在数学.LO/9910159中，我们通过构造一大类分裂器来回答一个公开问题。经典分裂器是自由模和无挠的代数紧模。在数学.LO/9910159中，我们专注于比连续统更大的分裂器，并且可数子模不一定是自由的。“相反”的情况下aleph_1-自由分裂的基数小于或等于aleph_1被挑出来，因为基本上不同的技术。这就是本文的研究目标。如果分裂子是可数的，那么根据Hausen的一个结果，它在有理数的某个子环上是自由的。我们可以证明基数aleph_1的所有aleph_1-free分裂器确实是自由的。
英文摘要：
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.
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