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

http://dx.doi.org/10.4153/CJM-2003-032-8
Canad. J. Math. 55(2003), 750-765

  Published:2003-08-01
 Printed: Aug 2003

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 $\ZFC$.