Absolutely indecomposable modules

Göbel, Rüdiger; Shelah, Saharon

doi:10.1090/s0002-9939-07-08725-4

Cited by 11 publications

(3 citation statements)

References 21 publications

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“…These kind of problems were considered by Fuchs, Göbel, Shelah and others (see e.g. [7,10,11]), probably the most important problem in this area is the problem of existence of absolutely indecomposable groups in every cardinality which remains open to this day (despite several partial answers are known).…”

Section: Introductionmentioning

confidence: 99%

On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups

Paolini¹,

Shelah²

2021

Preprint

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We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups G of size < p with infinite Torp(G), and that in particular there are no infinite reduced abelian p-groups of size < p. Secondly, we prove that if 2 ℵ 0 < λ < λ ℵ 0 , and G is abelian of size λ, then G is not co-Hopfian. Finally, we prove that for every cardinal λ there is a torsion-free abelian group G of size λ which is absolutely Hopfian, i.e., G is Hopfian and G remains Hopfian in every forcing extensions of the universe.

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Section: Introductionmentioning

confidence: 99%

On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups

Paolini¹,

Shelah²

2021

Preprint

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“…Thus it is reasonable for problems concerning absolute properties of graphs to restrict to objects of size < κ(ω). Fortunately we have a parallel result on absolute R 4 -modules derived from [12,9]; see details in [13,Chapter 22,Vol. 2], which matches to graphs of size < κ(ω).…”

Section: An Appendix On Absolute Constructionsmentioning

confidence: 99%

An axiomatic construction of an almost full embedding of the category of graphs into the category of -objects

Göbel

Przeździecki

2014

Journal of Pure and Applied Algebra

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We construct embeddings G of the category of graphs into categories of Rmodules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of Gare isomorphisms. The symbol R[S] above denotes the free R-module with the basis S. This implies that, for any cotorsion-free ring R, the categories of Rmodules are not less complicated than the category of graphs. A similar embedding of graphs into the category of vector spaces with four distinguished subspaces (over any field, e.g. F 2 = {0, 1}) is obtained.

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“…This notion, with still a different name, appears in work of Göbel-Shelah[2] and is related to Hjorth's notion of an "eplag" in[3].…”

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confidence: 99%