Almost-Free <i>E</i>-Rings of Cardinality ℵ<sub>1</sub>

Göbel, Rüdiger; Shelah, Saharon; Strüngmann, Lutz

doi:10.4153/cjm-2003-032-8

Can. j. math.

2003

DOI: 10.4153/cjm-2003-032-8

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Almost-Free E-Rings of Cardinality ℵ₁

Rüdiger Göbel¹,

Saharon Shelah²,

Lutz Strüngmann³

Abstract: 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 ℵ 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 ℵ 1 ≤ λ ≤ 2 ℵ 0 we construct E-rings of cardinality λ in ZFC which have ℵ 1 -free additive structure. For λ = ℵ 1 we therefore obtain the existence of almost-free E-ri… Show more

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“…Let us point out that Göbel-Shelah-Strüngmann [10] proves the existence of ℵ 1 -free E-rings of cardinality ℵ 1 .…”

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confidence: 99%

Large superdecomposable E(R)-algebras

Fuchs

Göbel

2005

Fund. Math.

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Large superdecomposable E(R)-algebras by Laszlo Fuchs (New Orleans) and Rüdiger Göbel (Essen)In honour of Claus Michael Ringel on the occasion of his 60th birthday Abstract. For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands = 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module A is multiplication by an element of A.

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“…Let us point out that Göbel-Shelah-Strüngmann [10] proves the existence of ℵ 1 -free E-rings of cardinality ℵ 1 .…”

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confidence: 99%

Large superdecomposable E(R)-algebras

Fuchs

Göbel

2005

Fund. Math.

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“…E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3,5,7,8,10,13,14,15,18,19]. Despite some efforts ([14, 5]) it remained an open question whether proper generalized E(R)-algebras exist.…”

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confidence: 99%

Generalized E-algebras via λ-calculus I

Göbel

Shelah

2006

Fund. Math.

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Abstract. An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra End R A of the R-module R A, taking any a ∈ A to the right multiplication a r ∈ End R A by a, is an isomorphism of algebras. In this case R A is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3,5,7,8,10,13,14,15,18,19]. Despite some efforts ([14, 5]) it remained an open question whether proper generalized E(R)-algebras exist. These are R-algebras A isomorphic to End R A but not under the above canonical isomorphism, so not E(R)-algebras. This question was raised about 30 years ago (for R = Z) by Schultz [21] (see also Vinsonhaler [24]). It originates from Problem 45 in Fuchs [9], that asks for a characterization of the rings A for which A ∼ = End Z A (as rings). We answer Schultz's question, thus contributing a large class of rings for Fuchs' Problem 45 which are not E-rings. Let R be a commutative ring with an element p ∈ R such that the additive group R + is p-torsion-free and p-reduced (equivalently p is not a zero-divisor and n∈ω p n R = 0). As explained in the introduction we assume that either |R| < 2 ℵ 0 or R + is free (see Definition 1.1). The main tool is an interesting connection between λ-calculus (used in theoretical computer science) and algebra. It seems reasonable to divide the work into two parts; in this paper we work in V = L (Gödel's universe) where stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper [12]. However the general strategy will be the same, but the combinatorial arguments will utilize a prediction principle that holds under ZFC.

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Almost-Free E-Rings of Cardinality ℵ₁

Cited by 2 publications

References 13 publications

Large superdecomposable E(R)-algebras

Large superdecomposable E(R)-algebras

Generalized E-algebras via λ-calculus I

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Almost-Free E-Rings of Cardinality ℵ1

Cited by 2 publications

References 13 publications

Large superdecomposable E(R)-algebras

Large superdecomposable E(R)-algebras

Generalized E-algebras via λ-calculus I

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Almost-Free E-Rings of Cardinality ℵ₁