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 commutator full transitivity and strongly commutator full transitivity. We establish, inter alia, that these notions are strictly stronger than the classical concept of full transitivity, but there are nonetheless many parallels between these things.
The authors would like to express their sincere gratitude to the referee for his/her careful reading of the manuscript and the competent suggestions made.
Correction added after online publication 21 May 2015: In the second line of the proof of Lemma 3.12 the text passage has been added from “and that the basic subgroup B is” to “the pursued claim”. In the second line of the statement of Proposition 3.16 the text “and vice versa” has been added after “then G is a cft-group”. The last sentence of the proof of Proposition 3.16 has been added.
© 2015 by De Gruyter