Generalized E-Rings

Rüdiger Göbel; Saharon Shelah; Lutz Strüngmann

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arXiv · math/0404271

Generalized E-Rings

Rüdiger Göbel·Saharon Shelah·Lutz Strüngmann

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Abstract

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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BibTeX RIS

Rüdiger Göbel, Saharon Shelah, Lutz Strüngmann. 2004-04-15. Generalized E-Rings. https://arxiv.org/abs/math/0404271

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