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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