### 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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