Almost-Free E-Rings of Cardinality aleph1

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 2aleph0 is undecidable in \ZFC. While they exist in Giodelʹs universe, they do not exist in other models of set theory. For a regular cardinal aleph1 N (lambda) N 2aleph0 we construct E-rings of cardinality (lambda) in \ZFC which have aleph1-free additive structure. For lambda = aleph1 we therefore obtain the existence of almost-free E-rings of cardinality aleph1 in ZFC.