Almost-free E-rings of cardinality aleph_1
Logic
2007-05-23 v1 Rings and Algebras
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^{aleph_0} 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 aleph_1 <= lambda <= 2^{aleph_0} we construct E-rings of cardinality lambda in ZFC which have aleph_1-free additive structure. For lambda = aleph_1 we therefore obtain the existence of almost-free E-rings of cardinality aleph_1 in ZFC.
Keywords
Cite
@article{arxiv.math/0112214,
title = {Almost-free E-rings of cardinality aleph_1},
author = {Rüdiger Göbel and Saharon Shelah and Lutz Strüngmann},
journal= {arXiv preprint arXiv:math/0112214},
year = {2007}
}