Centrally Essential Endomorphism Rings of Abelian Groups

We study Abelian groups $A$ with centrally essential endomorphism ring $\text{End}\,A$. If $A$ is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\,A$ is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings. The paper will appear in Communications in Algebra.