### 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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