Almost-free E-rings of cardinality aleph₁

2001/12/20 by Rüdiger Göbel, Saharon Shelah, Göbel, Rüdiger +3
Mathematics · #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA) #math.LO #math.RA

paper · pdf · doi:10.48550/arxiv.math/0112214

published as Canad. J. Math. 55 No. 4 (2003) 750--765

arxiv created 2001/12/20 · arxiv updated 2009/11/30

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^aleph₀ 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 aleph₁ <= lambda <= 2^aleph₀ we construct E-rings of cardinality lambda in ZFC which have aleph₁-free additive structure. For lambda = aleph₁ we therefore obtain the existence of almost-free E-rings of cardinality aleph₁ in ZFC.

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