Generalized E-Rings
نویسندگان
چکیده
A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End (RZ) of the additive group RZ, taking any r ∈ R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case RZ is called an E-group. Obvious examples of E-rings are subrings of Q. However there is a proper class of examples constructed recently, see [8]. E-rings come up naturally in various topics of algebra, see the introduction. So its not surprising that they were investigated thoroughly in the last decade, see [7, 21, 4, 10, 18]. 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, see [14]. 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 (RZ) ∼= R but R is not an E-ring.
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