Two-sided E-rings. (English) Zbl 1034.20048
The main objective of this paper is to solve a long-standing problem in Abelian group theory, namely to construct non-commutative rings \(R\) for which \(R\) is isomorphic to its ring \(\text{End}(R^+)\) of additive endomorphisms. (In the commutative case, such rings are called E-rings and they can be characterised as those rings for which \(\text{End}(R^+)\) is the ring of multiplications by \(R\).) The authors solve the problem by the simple yet ingenious idea of considering rings \(R\), called two-sided E-rings, for which \(\text{End}(R^+)\) is generated by its subring of left multiplications and its subring of right multiplications.
In the finite rank case, the authors characterise the rational algebra generated by left and right multiplications from \(R\). They finally use Shelah’s Black Box method to construct large two-sided E-rings.
In the finite rank case, the authors characterise the rational algebra generated by left and right multiplications from \(R\). They finally use Shelah’s Black Box method to construct large two-sided E-rings.
Reviewer: Phillip Schultz (Nedlands)
MSC:
| 20K20 | Torsion-free groups, infinite rank |
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 16S50 | Endomorphism rings; matrix rings |
Keywords:
left and right multiplications in rings; quasi-two-sided E-rings; torsion-free Abelian groups; additive endomorphisms; Shelah’s Black BoxReferences:
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