Abelian groups with enough $\pi$-regular ring of endomorphisms

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 L. Fuchs and K. 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. In the present paper we study the abelian groups with enough $\pi$-regular endomorphism ring (i. e., a subclass of abelian groups with $\pi$-regular endomorphism ring) and find the necessary and sufficient conditions for arbitrary abelian groups with enough $\pi$-regular endomorphism ring.