Large E-rings exist
type: Item
instance of: scholarly article
title: Large E-rings exist
zbMATH Open document ID: 0616.20026
DOI: 10.1016/0021-8693(87)90123-2
author: Manfred Dugas
author: Adolf Mader
author: Charles Vinsonhaler
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: A ring R with 1 is called an E-ring provided \(Hom_{{\mathbb{Z}}}(R,R)=R\) under the map \(\phi\to 1\phi\). E-rings were defined by \textit{P. Schultz} [J. Aust. Math. Soc. 15, 60-69 (1973; Zbl 0257.20037)]. They are of interest for the ''theory'' of torsion-free abelian groups, mainly in case of groups of finite rank. This connection was observed in earlier work by \textit{R. S. Pierce} [Mich. Math. J. 7, 241-243 (1960; Zbl 0103.268)]. Obvious examples of E-rings are subrings of the rationals. The only classical examples are quite similar, which restricts their size drastically \((\leq 2^{\aleph_ 0})\). In this paper E-rings are constructed for all infinite cardinals \(\lambda^{\aleph_ 0}\). The construction is based on the notion cotorsion-free and a combinatorial method, ''Shelah's Black Box'' which replaces additional set theoretic assumptions as the ''Jensen functions'' [see \textit{A. L. S. Corner}, \textit{R. Göbel}, Proc. Lond. Math. Soc., III. Ser. 50, 447-479 (1985; Zbl 0562.20030)]. The interesting construction is similar to recent constructions of indecomposable abelian groups.
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 3998091
zbMATH Keywords: E-rings
zbMATH Keywords: torsion-free abelian groups
zbMATH Keywords: infinite cardinals
zbMATH Keywords: cotorsion-free
zbMATH Keywords: Black Box
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90123-2
OpenAlex ID: W2017122272
cites work: Torsion-free Abelian groups of finite rank projective as modules over their endomorphism rings
cites work: Torsion-free rings
cites work: Q5726743
cites work: Unital rings whose additive endomorphisms commute
cites work: Prescribing Endomorphism Algebras, a Unified Treatment
cites work: Each countable reduced torsion-free commutative ring is a pure subring of an e-ring
cites work: Structure of categories
cites work: Abelian groups projective over their endomorphism rings
cites work: Subrings of simple algebras
cites work: Q3661679
cites work: The endomorphism ring of the additive group of a ring
Recommended article: Q3819209
Recommended article: Q2948292
Recommended article: Black rings at large \(D\)
Recommended article: Large fpf rings
Recommended article: Charged black rings at large D
Recommended article: Large universal deformation rings
Recommended article: Orbital perturbations due to massive rings
Recommended article: Mechanics of cosmic rings
Recommended article: Classifying E-Rings
Recommended article: On large fpf-rings
description: scientific article; zbMATH DE number 3998091