Abelian Groups With Monomorphisms Invariant With Respect to Epimorphisms

If for any injective endomorphism and surjective endomorphism of an abelian group there exists its endomorphism such that = (respectively, = ), then we say that the group possesses the R-property (respectively, the L-property). We show that if a reduced torsionfree group possesses the R-property or the L-property, then the endomorphism ring of the group is normal. We describe divisible groups and direct sums of cyclic groups possessing the R-property or the L-property.