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# Almost-Free $E$-Rings of Cardinality $\aleph_1$

Published:2003-08-01
Printed: Aug 2003
• Rüdiger Göbel
• Saharon Shelah
• Lutz Strüngmann
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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 G\"odel's universe, they do not exist in other models of set theory. For a regular cardinal $\aleph_1 \leq \lambda \leq 2^{\aleph_0}$ we construct $E$-rings of cardinality $\lambda$ in $\ZFC$ which have $\aleph_1$-free additive structure. For $\lambda=\aleph_1$ we therefore obtain the existence of almost-free $E$-rings of cardinality $\aleph_1$ in $\ZFC$.
 Keywords: $E$-rings, almost-free modules
 MSC Classifications: 20K20 - Torsion-free groups, infinite rank 20K30 - Automorphisms, homomorphisms, endomorphisms, etc. 13B10 - Morphisms 13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]

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