Almost-free E-rings of cardinality א1

Abstract

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 א0 is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal א1 ≤ λ ≤ 2א0 we construct E-rings of cardinality λ in ZFC which have א1 -free additive structure. For λ = א1 we therefore obtain the existence of almost-free E-rings of cardinality א1 in ZFC.