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Almost-Free E-Rings of Cardinality ℵ1

Published online by Cambridge University Press:  20 November 2018

Rüdiger Göbel
Affiliation:
Fachbereich 6—Mathematik, University of Essen, 45117 Essen, Germany e-mail: R.Goebel@uni-essen.de
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel e-mail: shelah@math.huji.ac.il lutz@math.huji.ac.il
Lutz Strüngmann
Affiliation:
Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel e-mail: shelah@math.huji.ac.il lutz@math.huji.ac.il
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Abstract

An $E$ -ring is a unital ring $R$ such that every endomorphism of the underlying abelian group ${{R}^{+}}$ is multiplication by some ring element. The existence of almost-free $E$ -rings of cardinality greater than ${{2}^{{{\aleph }_{0}}}}$ is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ${{\aleph }_{1}}\le \text{ }\!\!\lambda\!\!\text{ }\le {{2}^{{{\aleph }_{0}}}}$ we construct $E$ -rings of cardinality $\lambda $ in ZFC which have ${{\aleph }_{1}}$ -free additive structure. For $\text{ }\!\!\lambda\!\!\text{ }={{\aleph }_{1}}$ we therefore obtain the existence of almost-free $E$ -rings of cardinality ${{\aleph }_{1}}$ in ZFC.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2003

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