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Generalized E-Rings | | Rüdiger Göbel
; Saharon Shelah
; Lutz Strüngmann
; | | Date: |
15 Apr 2004 | | Subject: | Logic; Group Theory; Rings and Algebras | math.LO math.GR math.RA | | 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. | | Source: | arXiv, math.LO/0404271 | | Services: | Forum | Review | PDF | Favorites | |
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