Abelian groups with $\pi$-regular center of the endomorphism ring

The concept of $\pi$-regular endomorphism ring of an abelian group, which generalizes the concept of regular endomorphism ring was introduced in the works of Fuchs and Rangaswamy. They described a periodic abelian groups with $\pi$-regular endomorphism rings and found some necessary conditions for an abelian group with $\pi$-regular endomorphism rings. The abelian groups with $\pi$-regular center of the endomorphism ring are studied in the present paper. The necessary and sufficient conditions are found under which an arbitrary abelian group has a $\pi$-regular center of the endomorphism ring.