Endomorphism of Abelian Groups as Modules over Their Endomorphism Rings

For an Abelian group $$A$$ , viewed as a module over its endomorphism ring $$E(A)$$ , the near-ring $$\mathcal{M}_{E(A)}(A)$$ of homogeneous mappings is defined as the set of mappings $$\{f\colon A\to A \mid f(\varphi a)=\varphi f(a)$$ for all $$\varphi\in E(A)$$ and $$a\in A\}$$ with the operations of addition and composition (as multiplication). It is proved that the problem of describing some classes of mixed Abelian groups with the property $$\mathcal{M}_{E(A)}(A)=E(A)$$ reduces to the cause of torsion-free Abelian groups. Abelian groups with this property are found in the class of strongly indecomposable torsion-free Abelian groups of finite rank and torsion-free Abelian groups of finite rank coinciding with their pseudosocle.