Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group $A$ with a normal endomorphism ring contains a pure fully invariant subgroup $G\oplus B$, the endomorphism ring of a group $G$ is commutative, and a subgroup $B$ is not always distinguished by a direct summand in $A$. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.