Prescribing endomorphism algebras of $\aleph_n$-free modules

Rüdiger Göbel   ; Daniel Herden   ; Saharon Shelah

Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1775-1816

Résumé

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass and Eklof, both jointly with Shelah. In the first case the authors consider almost-free abelian groups and assume the existence of large canonical, free subgroups. Nevertheless, there exist א1​-separable torsion-free groups G of size א1​ with a basic subgroup B of rank א1​ such that all subgroups of G disjoint from B are also free, but the groups G are still not free. What else can we say about G? The other paper deals with Kaplansky's test problems (which are excellent indicators that the objects defy classification). The authors are able to construct very free abelian groups and verify the test problems for them by a careful choice of \emph{particular} elements of their endomorphism rings.

Détail

DOI : 10.4171/jems/475

Classification : 16-XX, 00-XX, 01-XX
Keywords: Prediction principles, almost free abelian groups, endomorphism rings