Divisible Groups

Fuchs, L.

doi:10.1016/b978-0-08-009206-5.50007-8

Cited by 48 publications

(103 citation statements)

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“…To see this, choose a subgroup ≤ R such that ∼ = R and (R : ) = 2 ℵ 0 (such exists by the structure theorem for divisible Abelian groups; cf., for instance, [4,Theorem 19.1]). Let {f σ } σ ∈S be a family of pairwise locally incompatible test functions (with bijective indexing), such that C RF (G) (f σ ) ∼ = for all σ ∈ S, and such that |S| = |G| (R: ) = c G (such a family exists by [7,Theorem 30]).…”

Section: Corollary 54mentioning

confidence: 97%

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Müller

2010

Abh. Math. Semin. Univ. Hambg.

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This paper continues the investigation of the groups RF (G) first introduced in the forthcoming book of Chiswell and Müller "A Class of Groups Universal for Free R-Tree Actions" and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, 2009). We establish a criterion for a family {H σ } of hyperbolic subgroups H σ ≤ RF (G) to generate a hyperbolic subgroup isomorphic to the free product of the H σ (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in RF (G) for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in RF (G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).

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Section: Corollary 54mentioning

confidence: 97%

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Müller

2010

Abh. Math. Semin. Univ. Hambg.

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“…We write N ⊆ c X M . The notion of X -closed submodule was used by many authors under different names; let us recall some of them: X -pure submodule, by J. S. Golan in [5], Xisolated submodule, by T. H. Fay and S. V. Joubert in [3], X -super-honest submodule, by S. V. Joubert and M. J. Schoeman in [8] (the last two when X is a set of elements of R); and, of course, we have the corresponding notion when R = and X is the set of all nonzero integer numbers, see [4].…”

Section: Closed Submodulesmentioning

confidence: 98%

Honest submodules

Jara

2007

Czech Math J

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Lattices of submodules of modules and the operators we can define on these lattices are useful tools in the study of rings and modules and their properties. Here we shall consider some submodule operators defined by sets of left ideals. First we focus our attention on the relationship between properties of a set of ideals and properties of a submodule operator it defines. Our second goal will be to apply these results to the study of the structure of certain classes of rings and modules. In particular some applications to the study and the structure theory of torsion modules are provided.

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“…Clearly it suffices to prove (2) implies (3). Assume [f, g] has finite p-height in End (G) for some p ~ S. Then a suitable multiple has p-height zero; since G ~ Gp 9 H v, say, where Hp = {x E G:h(x)(p) = ~} there exists a E Gp such that 0r g](a).…”

Section: ) If F Geg(g) Then the Homomorphism F: U/t ~ G Induced Bymentioning

confidence: 99%

On a paper of Szele and Szendrei on groups with commutative endomorphism rings

Schultz

1973

Acta Mathematica Academiae Scientiarum Hungaricae

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Divisible Groups

Cited by 48 publications

References 0 publications

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Honest submodules

On a paper of Szele and Szendrei on groups with commutative endomorphism rings

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