Abstract view
AlmostFree $E$Rings of Cardinality $\aleph_1$


Published:20030801
Printed: Aug 2003
Rüdiger Göbel
Saharon Shelah
Lutz Strüngmann
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 almostfree $E$rings of
cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$.
While they exist in G\"odel'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
almostfree $E$rings of cardinality $\aleph_1$ in $\ZFC$.
MSC Classifications: 
20K20, 20K30, 13B10, 13B25 show english descriptions
Torsionfree groups, infinite rank Automorphisms, homomorphisms, endomorphisms, etc. Morphisms Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
20K20  Torsionfree groups, infinite rank 20K30  Automorphisms, homomorphisms, endomorphisms, etc. 13B10  Morphisms 13B25  Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
