# Sh:682

- Göbel, R., & Shelah, S. (1999).
*Almost free splitters*. Colloq. Math.,**81**(2), 193–221. arXiv: math/9910161 DOI: 10.4064/cm-81-2-193-221 MR: 1715347

See [Sh:E22] -
Abstract:

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is {\rm Ext}_R(G,G)=0 holds. For simplicity we will call such modules splitters. Our investigation continues [GbSh:647]. In [GbSh:647] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [GbSh:647] 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. - published version (29p)

Bib entry

@article{Sh:682, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{Almost free splitters}}, journal = {Colloq. Math.}, fjournal = {Colloquium Mathematicum}, volume = {81}, number = {2}, year = {1999}, pages = {193--221}, issn = {0010-1354}, mrnumber = {1715347}, mrclass = {20K35 (13D07 18G15 20K20)}, doi = {10.4064/cm-81-2-193-221}, note = {\href{https://arxiv.org/abs/math/9910161}{arXiv: math/9910161}}, arxiv_number = {math/9910161}, keyword = {self-splitting modules, criteria for freeness of modules}, referred_from_entry = {See [Sh:E22]} }