Geodesic
Almost free splitters
Colloquium Mathematicum, Tome 81 (1999) no. 2, pp. 193-221.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ext_R(G,G) = 0$. 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 $ℵ_1$-free splitters of cardinality less than or equal to $ℵ_1$ 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 $ℵ_1$-free splitters of cardinality $ℵ_1$ are free indeed.
DOI : 10.4064/cm-81-2-193-221
Keywords: self-splitting modules, criteria for freeness of modules

Rüdiger Göbel 1 ; Saharon Shelah 1

1 
@article{10_4064_cm_81_2_193_221,
     author = {R\"udiger G\"obel and Saharon Shelah},
     title = {Almost free splitters},
     journal = {Colloquium Mathematicum},
     pages = {193--221},
     publisher = {mathdoc},
     volume = {81},
     number = {2},
     year = {1999},
     doi = {10.4064/cm-81-2-193-221},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-81-2-193-221/}
}
TY  - JOUR
AU  - Rüdiger Göbel
AU  - Saharon Shelah
TI  - Almost free splitters
JO  - Colloquium Mathematicum
PY  - 1999
SP  - 193
EP  - 221
VL  - 81
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4064/cm-81-2-193-221/
DO  - 10.4064/cm-81-2-193-221
LA  - en
ID  - 10_4064_cm_81_2_193_221
ER  - 
%0 Journal Article
%A Rüdiger Göbel
%A Saharon Shelah
%T Almost free splitters
%J Colloquium Mathematicum
%D 1999
%P 193-221
%V 81
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4064/cm-81-2-193-221/
%R 10.4064/cm-81-2-193-221
%G en
%F 10_4064_cm_81_2_193_221
Rüdiger Göbel; Saharon Shelah. Almost free splitters. Colloquium Mathematicum, Tome 81 (1999) no. 2, pp. 193-221. doi : 10.4064/cm-81-2-193-221. http://geodesic.mathdoc.fr/articles/10.4064/cm-81-2-193-221/

Cité par Sources :