Almost-free E-rings of cardinality aleph_1

. 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 ℵ 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 ℵ 1 ≤ λ ≤ 2 ℵ 0 we construct E -rings of cardinality λ in ZFC which have ℵ 1 -free additive structure. For λ = ℵ 1 we therefore obtain the existence of almost-free E -rings of cardinality ℵ 1 in ZFC.