  A ring R is called an E-ring if the canonical homomorphism from R to the
endomorphism ring End(R_Z) of the additive group R_Z, taking any r in R to the
endomorphism left multiplication by r turns out to be an isomorphism of rings.
In this case R_Z is called an E-group. Obvious examples of E-rings are subrings
of Q. However there is a proper class of examples constructed recently. E-rings
come up naturally in various topics of algebra. This also led to a
generalization: an abelian group G is an E-group if there is an epimorphism
from G onto the additive group of End(G). If G is torsion-free of finite rank,
then G is an E-group if and only if it is an E-group. The obvious question was
raised a few years ago which we will answer by showing that the two notions do
not coincide. We will apply combinatorial machinery to non-commutative rings to
produce an abelian group G with (non-commutative) End(G) and the desired
epimorphism with prescribed kernel H. Hence, if we let H=0, we obtain a
non-commutative ring R such that End(R_{Z}) cong R but R is not an E-ring.
