Almost-free E-rings of cardinality א1

Rüdiger Göbel; Saharon Shelah; Lutz Strüngmann

doi:10.4153/CJM-2003-032-8

Almost-free E-rings of cardinality א₁

Rüdiger Göbel^*
, Saharon Shelah
, Lutz Strüngmann

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 ^א0 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 א₁ ≤ λ ≤ 2^א0 we construct E-rings of cardinality λ in ZFC which have א₁ -free additive structure. For λ = א₁ we therefore obtain the existence of almost-free E-rings of cardinality א₁ in ZFC.

Original language	English
Pages (from-to)	750-765
Number of pages	16
Journal	Canadian Journal of Mathematics
Volume	55
Issue number	4
DOIs	https://doi.org/10.4153/CJM-2003-032-8
State	Published - Aug 2003

Keywords

Almost-free modules
E-rings

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10.4153/CJM-2003-032-8

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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 א0 is undecidable in ZFC. While they exist in G{\"o}del's universe, they do not exist in other models of set theory. For a regular cardinal א1 ≤ λ ≤ 2א0 we construct E-rings of cardinality λ in ZFC which have א1 -free additive structure. For λ = א1 we therefore obtain the existence of almost-free E-rings of cardinality א1 in ZFC.",

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N2 - 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 א0 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 א1 ≤ λ ≤ 2א0 we construct E-rings of cardinality λ in ZFC which have א1 -free additive structure. For λ = א1 we therefore obtain the existence of almost-free E-rings of cardinality א1 in ZFC.

AB - 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 א0 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 א1 ≤ λ ≤ 2א0 we construct E-rings of cardinality λ in ZFC which have א1 -free additive structure. For λ = א1 we therefore obtain the existence of almost-free E-rings of cardinality א1 in ZFC.

KW - Almost-free modules

KW - E-rings

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JF - Canadian Journal of Mathematics

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ER -

Almost-free E-rings of cardinality א₁

Abstract

Keywords

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