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

by Goebel and Shelah and Struengmann. [GShS:785]
Canadian J Math, 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 Goedel's universe, they do not exist in other models of set theory. For a regular cardinal aleph_1 <= lambda <= 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.

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