Abelian groups with monomorphisms invariant with respect to epimorphisms

If for any injective endomorphism $\alpha$ and surjective endomorphism $\beta$ of abelian group there exist its endomorphism $\gamma$ such that $\beta\alpha=\alpha\gamma$ ($\alpha\beta=\gamma\alpha$, respectively), then such a property of the group is called $R$-property ($L$-property, respectively). It is shown that if reduced torsion-free group possesses $R$- or $L$-property, then endomorphism ring of a group is normal. We describe the divisible groups and direct sums of cyclic groups with $R$- or $L$-property.