# Sh:681

• Göbel, R., Shelah, S., & Strüngmann, L. H. (2004). Generalized E-rings. In Rings, modules, algebras, and abelian groups, Vol. 236, Dekker, New York, pp. 291–306.
• Abstract:
A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_{\mathbb Z}) of the additive group R_{\mathbb 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_{\mathbb Z} is called an E-group. Obvious examples of E-rings are subrings of {\mathbb 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 {\mathbb 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 {\mathbb 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_{{\mathbb Z}}) \cong R but R is not an E-ring.
Bib entry
@incollection{Sh:681,
author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon and Str{\"u}ngmann, Lutz H.},
title = {{Generalized $E$-rings}},
booktitle = {{Rings, modules, algebras, and abelian groups}},
series = {Lecture Notes in Pure and Appl. Math.},
volume = {236},
year = {2004},
pages = {291--306},
mrclass = {20K30 (20K20)},
mrnumber = {2050718},
mrreviewer = {B. Goldsmith},
publisher = {Dekker, New York},
note = {\href{https://arxiv.org/abs/math/0404271}{arXiv: math/0404271}},
arxiv_number = {math/0404271}
}