Abelian groups with left morphic endomorphism ring

The main purpose of this paper is to determine as much as possible, large classes of Abelian groups which have left morphic endomorphism ring. We show that morphic Abelian groups and Abelian groups whose endomorphism ring is left (or right) morphic coincide in the torsion, divisible, torsion-free and splitting mixed cases. In some cases, we obtain more general results concerning image-projective (or image-injective) Abelian groups.