Absolute $E$-rings

by Goebel and Herden and Shelah. [GbHeSh:948]
Advances in Math, 2011
A ring R with 1 is called an E-ring if End_Z R is ring-isomorphic to R under the canonical homomorphism taking the value 1 sigma for any sigma in End_Z R . 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 lambda^{aleph_0} was shown by Dugas, Mader and Vinsonhaler [DMV]. We want to show the existence of absolute E-rings. It turns out that there is a precise cardinal-barrier kappa (omega) for this problem: (The cardinal kappa (omega) is the first omega-Erdos cardinal defined in the introduction. It is a relative of measurable cardinals.) We will construct absolute E-rings of any size lambda < kappa (omega) . But there are no absolute E-rings of cardinality >= kappa (omega) . The non-existence of huge, absolute E-rings >= kappa (omega) follows from a recent paper by Herden and Shelah [HS] and the construction of absolute E-rings R is based on an old result by Shelah [S] 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 lambda < kappa (omega), 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 ([DMV,GT]) of cardinality >= 2^{aleph_0} 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 lambda^{aleph_0}). These E-rings are domains and as a by-product we obtain the existence of absolutely indecomposable abelian groups, compare [GS2].

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