On Projectively Fully Transitive Abelian p-Groups

Abstract

There are two natural questions which arise in connection with the endomorphism ring of an Abelian group: when is the ring generated by its idempotents and when is the ring generated additively by its idempotents? The present work investigates these two questions for Abelian p-groups. This leads in a natural way to consideration of two strengthened versions of Kaplansky’s notion of full transitivity, which we call projective full transitivity and strong projective full transitivity. We establish, inter alia, that these concepts are strictly stronger than the classical concept of full transitivity but there are nonetheless many strong parallels between the notions.