Almost-Free E-Rings of Cardinality א1

Göbel, Rüdiger ; Shelah, Saharon ; Strüngmann, Lutz

Canadian journal of mathematics, Tome 55 (2003) no. 4, pp. 750-765

Résumé

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 Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ${{\aleph }_{1}}\le \text{ }\!\!\lambda\!\!\text{ }\le {{2}^{{{\aleph }_{0}}}}$ we construct $E$ -rings of cardinality $\lambda $ in ZFC which have ${{\aleph }_{1}}$ -free additive structure. For $\text{ }\!\!\lambda\!\!\text{ }={{\aleph }_{1}}$ we therefore obtain the existence of almost-free $E$ -rings of cardinality ${{\aleph }_{1}}$ in ZFC.

Détail

DOI : 10.4153/CJM-2003-032-8

Mots-clés : 20K20, 20K30, 13B10, 13B25, E-rings, almost-free modules

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