Absolute E-rings

A ring R with 1 is called an E-ring if EndZR is ring-isomorphic to R under the canonical homomorphism taking the value 1σ for any σ∈EndZR. Moreover R is an absolute E-ring if it remains an E-ring in every generic extension of the universe. E-rings are an important tool for algebraic topology as explained in the introduction. The existence of an E-ring R of each cardinality of the form λℵ₀ was shown by Dugas, Mader and Vinsonhaler (1987) 9]. We want to show the existence of absolute E-rings. It turns out that there is a precise cardinal-barrier κ(ω) for this problem: (The cardinal κ(ω) is the first ω-Erd?s cardinal defined in the introduction. It is a relative of measurable cardinals.) We will construct absolute E-rings of any size λ<κ(ω). But there are no absolute E-rings of cardinality ?κ(ω). The non-existence of huge, absolute E-rings ?κ(ω) follows from a recent paper by Herden and Shelah (2009) 24] and the construction of absolute E-rings R is based on an old result by Shelah (1982) 31] where families of absolute, rigid colored trees (with no automorphism between any distinct members) are constructed. We plant these trees into our potential E-rings with the aim to prevent unwanted endomorphisms of their additive group to survive. Endomorphisms will recognize the trees which will have branches infinitely often divisible by primes. Our main result provides the existence of absolute E-rings for all infinite cardinals λ<κ(ω), i.e. these E-rings remain E-rings in all generic extensions of the universe (e.g. using forcing arguments). Indeed all previously known E-rings (Dugas, Mader and Vinsonhaler, 1987 9]; Göbel and Trlifaj, 2006 23]) of cardinality ?ℵ₀² have a free additive group R⁺ in some extended universe, thus are no longer E-rings, as explained in the introduction. Our construction also fills all cardinal-gaps of the earlier constructions (which have only sizes λℵ₀). These E-rings are domains and as a by-product we obtain the existence of absolutely indecomposable abelian groups, compare Göbel and Shelah (2007) 22].