-Separable Groups and Kaplansky’s Test Problems

Thomé, Bernhard

doi:10.1515/form.1990.2.203

Cited by 27 publications

(5 citation statements)

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“…One can allow such a distinguished function ranges over other classes such as finite-range, countable-range, inessential range or even small homomorphism, and there are a lot of work trying to clarify such situations. As a short list, we may mention the papers Corner-Göbel [7], Dugas-Göbel [10], Corner [8], Thome [33] and Pirece [18].…”

Section: G?mentioning

confidence: 99%

Co-Hopfian and boundedly endo-rigid mixed groups

Asgharzadeh¹,

Golshani²,

Shelah³

2022

Preprint

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For a given cardinal λ and a torsion abelian group K of cardinality less than λ, we present, under some mild conditions (for example λ = λ ℵ 0 ), boundedly endo-rigid abelian group G of cardinality λ with Tor(G) = K. Essentially, we give a complete characterization of such pairs (K, λ). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existence problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups.In particular, we give a useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals λ > 2 ℵ 0 for which there is a co-Hopfian abelian group of size λ.Contents § 1. Introduction 2 § 2. Preliminaries 6 § 3. The ZFC construction of boundedly rigid mixed groups 10 § 4. Co-Hopfian and boundedly endo-rigid abelian groups 41

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Section: G?mentioning

confidence: 99%

Co-Hopfian and boundedly endo-rigid mixed groups

Asgharzadeh¹,

Golshani²,

Shelah³

2022

Preprint

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“…One can allow such a distinguished function ranges over other classes such as finite-range, countable-range, inessential range or even small homomorphism, and there are a lot of work trying to clarify such situations. As a short list, we may mention Corner and Göbel [8], Dugas and Göbel [10], Corner [7], Thomé [30] and Pierce [21].…”

Section: Introductionmentioning

confidence: 99%

Co-Hopfian and boundedly endo-rigid mixed abelian groups

Asgharzadeh,

Golshani,

Shelah

2023

Pacific J. Math.

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For a given cardinal λ and a torsion abelian group K of cardinality less than λ, we present, under some mild conditions (for example, λ = λ ℵ 0 ), boundedly endo-rigid abelian group G of cardinality λ with tor(G) = K. Essentially, we give a complete characterization of such pairs (K, λ). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existence problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals λ > 2 ℵ 0 for which there is a co-Hopfian abelian group of size λ. 1. Introduction 183 2. Preliminaries 187 3. The ZFC construction of boundedly rigid mixed groups 191 4. Co-Hopfian and boundedly endo-rigid abelian groups 219 Acknowledgement 230 References 231Golshani's research has been supported by a grant from IPM (no. 1401030417).

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“…Thomé [50] and Eklof-Shelah [15] constructed an ℵ 1 -separable Abelian group M in ZFC such that the Corner's ring is algebraically closed in End(M). Consequently M breaks down the Kaplansky's test problem, i.e., it is isomorphic to its cube but not to its square.…”

Section: Introductionmentioning

confidence: 99%

Kaplansky Test Problems for $R$-Modules in ZFC

Asgharzadeh,

Golshani,

Shelah

2021

Preprint

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Fix a ring R and look at the class of left R-modules and naturally we restrict ourselves to the case of rings such that this class is not too similar to the case R is a field.We shall solve Kaplansky test problems, all three of which say that we do not have decomposition theory, e.g., if the square of one module is isomorphic to the cube of another it does not follows that they are isomorphic. Our results are in the ordinary set theory, ZFC. For this we look at bimodules, i.e., the structures which are simultaneously left R-modules and right S-modules with reasonable associativity, over a commutative ring T included in the centers of R and of S. Eventually we shall choose S to help solve each of the test questions. But first we analyze what can be the smallest endomorphism ring of an (R, S)bimodule. We construct such bimodules by using a simple version of the black box.

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-Separable Groups and Kaplansky’s Test Problems

Cited by 27 publications

References 1 publication

Co-Hopfian and boundedly endo-rigid mixed groups

Co-Hopfian and boundedly endo-rigid mixed groups

Co-Hopfian and boundedly endo-rigid mixed abelian groups

Kaplansky Test Problems for $R$-Modules in ZFC

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