Abelian groups with a regular center of the endomorphism ring

The article is related to the following fundamental problem: "What rings are endomorphism rings of abelian groups?", as well as to the problem of describing abelian groups for which the center of endomorphism ring possesses one or another property. In particular, we consider abelian groups whose endomorphism rings are regular. L. Fuchs and K. M. Rangaswamy described abelian groups with a regular endomorphism ring by use of the reduced case. If a group is reduced and has a regular endomorphism ring, then its periodic part is elementary, its factor group is divisible, and the group is embedded in a direct product of the $p$-components of its periodic part, as is shown by those authors. Similar results were obtained for groups the centers of endomorphism rings of which are regular by A. V. Karpenko and V. M. Misyakov. In this article, some necessary and sufficient conditions for the existence of a regular endomorphism ring (of a regular center of the endomorphism ring) of a reduced Abelian group are found.