### Almost free splitters

by Goebel and Shelah. [GbSh:682]

Colloquium Math, 1999

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 [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.

Back to the list of publications