# Sh:785

- Göbel, R., Shelah, S., & Strüngmann, L. H. (2003).
*Almost-free E-rings of cardinality \aleph_1*. Canad. J. Math.,**55**(4), 750–765. arXiv: math/0112214 DOI: 10.4153/CJM-2003-032-8 MR: 1994072 -
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\leq\lambda\leq 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. - Version 2002-01-18_11 (17p) published version (16p)

Bib entry

@article{Sh:785, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon and Str{\"u}ngmann, Lutz H.}, title = {{Almost-free $E$-rings of cardinality $\aleph_1$}}, journal = {Canad. J. Math.}, fjournal = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, volume = {55}, number = {4}, year = {2003}, pages = {750--765}, issn = {0008-414X}, mrnumber = {1994072}, mrclass = {20K20 (13F10 20K30)}, doi = {10.4153/CJM-2003-032-8}, note = {\href{https://arxiv.org/abs/math/0112214}{arXiv: math/0112214}}, arxiv_number = {math/0112214} }