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1
Content available remote

An extension of Zassenhaus' theorem on endomorphism rings

100%
Dugas M., Göbel R.
Fundamenta Mathematicae
|
2007
|
tom 194
|
nr 3
239-251
EN
Let R be a ring with identity such that R⁺, the additive group of R, is torsion-free. If there is some R-module M such that $R ⊆ M ⊆ ℚR (= ℚ ⊗_{ℤ} R)$ and $End_{ℤ}(M) = R$, we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R⁺ is free of finite rank, then R is a Zassenhaus ring. We will show that if R⁺ is free of countable rank and each element of R is algebraic over ℚ, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A. L. S. Corner, answering Kaplansky's test problems in the negative for torsion-free abelian groups, is a Zassenhaus ring.
2
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Generalized E-algebras via λ-calculus I

100%
Göbel R., Shelah S.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 2
155-181
EN
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 $⋂_{n∈ω} pⁿR = 0$). As explained in the introduction we assume that either $|R| < 2^{ℵ₀}$ 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.
3
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Almost-free E(R)-algebras and E(A,R)-modules

100%
Göbel R., Strüngmann L.
Fundamenta Mathematicae
|
2001
|
tom 169
|
nr 2
175-192
EN
Let R be a unital commutative ring and A a unital R-algebra. We introduce the category of E(A,R)-modules which is a natural extension of the category of E-modules. The properties of E(A,R)-modules are studied; in particular we consider the subclass of E(R)-algebras. This subclass is of special interest since it coincides with the class of E-rings in the case R = ℤ. Assuming diamond ⋄, almost-free E(R)-algebras of cardinality κ are constructed for any regular non-weakly compact cardinal κ > ℵ ₀ and suitable R. The set-theoretic hypothesis can be weakened.
4
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Algebraic ramifications of the common extension problem for group-valued measures

100%
Göbel R., Shortt R. M.
Fundamenta Mathematicae
|
1994-1995
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tom 146
|
nr 1
1-20
EN
Let G be an Abelian group and let μ: A → G and ν: B → G be finitely additive measures (charges) defined on fields A and B of subsets of a set X. It is assumed that μ and ν agree on A ∩ B, i.e. they are consistent. The existence of common extensions of μ and ν is investigated, and conditions on A and B facilitating such extensions are given.
5
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Large superdecomposable E(R)-algebras

100%
Fuchs L., Göbel R.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 1
71-82
EN
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.
6
Content available remote

Subgroups of the Baer–Specker group with few endomorphisms but large dual

100%
Blass A., Göbel R.
Fundamenta Mathematicae
|
1996
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tom 149
|
nr 1
19-29
EN
Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group $ℤ^{ℵ_0}$ with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.
7
Content available remote

Embeddings of Kronecker modules into the category of prinjective modules and the endomorphism ring problem

100%
Göbel R., Simson D.
Colloquium Mathematicum
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1998
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tom 75
|
nr 2
213-244
8
Content available remote

An addendum and corrigendum to "Almost free splitters" (Colloq. Math. 81 (1999), 193-221)

100%
Göbel R., Shelah S.
Colloquium Mathematicum
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2001
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tom 88
|
nr 1
155-158
EN
Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if $Ext¹_{R}(G,G) = 0$. In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.
9
Content available remote

Almost free splitters

100%
Göbel R., Shelah S.
Colloquium Mathematicum
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1999
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tom 81
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nr 2
193-221
EN
Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ext_R(G,G) = 0$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel's paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger than the continuum and such that countable submodules are not necessarily free. The "opposite" case of $ℵ_1$-free splitters of cardinality less than or equal to $ℵ_1$ was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by Hausen [7]. In contrast to the results of [5] and in accordance with [7] we can show that all $ℵ_1$-free splitters of cardinality $ℵ_1$ are free indeed.
10
Content available remote

Endomorphism algebras over large domains

100%
Göbel R., Pabst S.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 3
211-240
EN
The paper deals with realizations of R-algebras A as endomorphism algebras End G ≅ A of suitable R-modules G over a commutative ring R. We are mainly interested in the case of R having "many prime ideals", such as R = ℝ[x], the ring of real polynomials, or R a non-discrete valuation domain
11
Content available remote

Cellular covers of cotorsion-free modules

81%
Göbel R., Rodríguez J., Strüngmann L.
Fundamenta Mathematicae
|
2012
|
tom 217
|
nr 3
211-231
EN
In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $π⁎: Hom_{R}(G,G) ≅ Hom_{R}(G,H)$, where π⁎(φ) = πφ for each $φ ∈ Hom_{R}(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free R-module of rank $κ < 2^{ℵ₀}$ is realizable as the kernel of some cellular cover G → H where the rank of G is 3κ + 1 (or 3, if κ = 1). The proof is based on Corner's classical idea of how to construct torsion-free abelian groups with prescribed countable endomorphism rings. This complements results by Buckner-Dugas. On the other hand, we prove that every cotorsion-free R-module H that satisfies some rigid conditions admits arbitrarily large cellular covers G → H. This improves results by Fuchs-Göbel and Farjoun-Göbel-Segev-Shelah.
12
Content available remote

Modules over arbitrary domains II

50%
Göbel R., Shelah S.
Fundamenta Mathematicae
|
1985-1986
|
tom 126
|
nr 3
217-243
13
Content available remote

Quotients of reflexive modules

50%
Dugas M., Göbel R.
Fundamenta Mathematicae
|
1981
|
tom 114
|
nr 1
17-28
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