On commutator fully transitive Abelian groups

Abstract

There are two rather natural questions which arise in connection with the endomorphism ring of an Abelian group: when is the ring generated by its commutators, and when is the ring additively generated by its commutators? The current work explores these two problems for arbitrary Abelian groups. This leads in a standard way to consideration of two improved versions of Kaplansky's notion of full transitivity, which we call