Localizations of torsion-free abelian groups II

Dugas, Manfred

doi:10.1016/j.jalgebra.2004.11.013

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…The main results of this section for the category mod-Z of Abelian groups are given in some form in [11,18].…”

Section: Preservation Of Ring or Module Structures By Idempotent Funcmentioning

confidence: 99%

“…Orthogonality, f -local objects, and f -equivalences have appeared in category theory; then they began to be considered for groups and modules (see [17,18,41,47]). Localization functors have a long history, and the functors were intensively studied in many fields of mathematics.…”

Section: Is a Localization If And Only If The S-module A ⊗ S R Is An mentioning

confidence: 99%

“…It follows from some result of Dugas [18] that not all localizations of torsion-free groups of rank 1 are of the form presented in Theorem 11.7. Dugas also constructed a torsion-free group C such that "the value" of the push-out from (4) is not isomorphic to L g C. Dugas writes that the problem of describing L g C remains unsolved.…”

Section: Corollary 112 Ifmentioning

confidence: 99%

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Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…The main results of this section for the category mod-Z of Abelian groups are given in some form in [11,18].…”

Section: Preservation Of Ring or Module Structures By Idempotent Funcmentioning

confidence: 99%

Section: Is a Localization If And Only If The S-module A ⊗ S R Is An mentioning

confidence: 99%

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Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…Recently, localizations in the category of abelian groups were studied in [4], [6], [7], [11], [15] and in other articles. We will use the following standard identifications (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…We will use the following standard identifications (see e.g. [6], [7], [9], [10]). If M is a faithful left R-module, then any r ∈ R will be identified with the scalar multiplication by r on the left.…”

Section: Introductionmentioning

confidence: 99%

Quasi-localizations of ℤ

Buckner

Dugas

2007

Isr. J. Math.

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Motivated by the categorical notion of localizations applied to the quasicategory of abelian groups, we call a homomorphism α: A → B a quasilocalization of abelian groups if for each ϕ ∈ Hom(A, B) there is an n ∈ N and a unique ψ ∈ End(B) such that nϕ = ψ • α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers Z. While it is well-known that localizations of Z are just the E-rings, quasi-localizations of Z are much more abundant; an injection α: Z → M with M torsion-free, is a quasi-localization if and only if, for R = End(M ), one has R ⊆ M ⊆ Q⊗ Z R. We call R the ring of the quasilocalization M . Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasilocalizations of Z. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R + , as is required here. We will use a particular case of a category of left R-modules M with a distinguished family F of submodules and thus End(M, F ) = {ψ ∈ End(M ) : ψ(X) ⊆ X for all X ∈ F}. We will restrict our discussion to the case M = R such that End(R, F ) = R, and in this case we call the family F of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of Z, among them those of infinite rank as well as matrix rings for various rings of finite rank.

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Co-local subgroups of abelian groups II

Dugas

2007

Journal of Pure and Applied Algebra

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In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] the notion of a colocal subgroup of an abelian group was introduced. A subgroup K of A is called co-local if the natural map Hom(A, A) → Hom(A, A/K ) is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups K of mixed, non-splitting abelian groups A with torsion subgroup t (A). We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D/t (A) is the divisible part of A/t (A) = D/t (A)⊕ H/t (A), then K ∩ D = 0, and one may assume that K ⊆ H . We will construct examples to show that K need not be a co-local subgroup of H . Moreover, we will investigate connections between co-local subgroups of A and A/t (A).

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Localizations of torsion-free abelian groups II

Cited by 5 publications

References 8 publications

Idempotent functors and localizations in categories of modules and Abelian groups

Idempotent functors and localizations in categories of modules and Abelian groups

Quasi-localizations of ℤ

Co-local subgroups of abelian groups II

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