### Generalized $E$-Rings

by Goebel and Shelah and Struengmann. [GShS:681]

Proc Venice Conference on Rings, modules, algebras, and abelian groups (2002); in the series: Lecture Notes in Pure and Appl. Math., 2004

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.

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