The endomorphism ring of pointed separable torsion-free Abelian groups

Abstract

This paper studies the endomorphism rings of separable torsion-free Abelian groups. Baer [l] defined an Abelian group to be separable if given any finite subset of the group there is a complctcly dccomposablc direct summand of the group containing that subset. The endomorphism rings of such groups have attracted the attention of several workers, notably that of Liebert [7] who characterized those rings isomorphic to the endomorphism ring of some separable Abelian p-group. Moreover, Baer [2] and Kaplansky [6] proved that if two torsion Abelian groups have isomorphic endomorphism rings, then the groups themselves are isomorphic. Results corresponding to these have been proven for homogeneous torsion- free Abelian groups, with Metelli and Salce [8] obtaining a characterization of the endomorphism rings of those groups and Hauptfleisch [5] showing that if two such groups have isomorphic endomorphism rings, then they are almost isomorphic. The meaning of “almost” becomes clear in Theorem 3, from which Hauptfleisch’s result follows as a corollary. We consider only torsion-free Abelian groups and any undefined concepts are standard ones from Fuchs [3]. We define a group to be pointed if the set of types of its rank-l summands contains a unique minimal type, when WC say that the group is pointed in that type. We say a group is idempointed if it is pointed in an idempotent type. Clearly homogeneous separable groups are pointed since all elements of the group have the same type and the group has at least one rank-l summand. We characterize the rings isomorphic to the endomorphism ring of some