Let R be an associative ring with a unit and TV be a left R-module. The set M_R(N) = {f: N → N | f(rx) -=rf(x), r ∈ R, x ∈N} is a near-ring with respect to the operations of addition and composition and contains the ring E_R(N) of all endomorphisms of the it-module N. The R-module N is endomorphic if M_R(N) = E_R(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.
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