Invariants for abelian groups and dual exact sequences

“…According to [8], a torsion-free group G of finite rank is said to be quotient divisible if G has a free Abelian subgroup F such that G/F is a divisible torsion group. Now quotient divisible groups are actively studied by various authors; e.g., see [1], [16], [20] and others.…”

Section: Homogeneous Murley Groupsmentioning

confidence: 99%

Rings on Abelian Torsion-Free Groups of Finite Rank

Kompantseva¹,

Tuganbaev²

2021

Preprint

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In the class of reduced Abelian torsion-free groups G of finite rank, we describe T I-groups, this means that every associative ring on G is filial. If every associative multiplication on G is the zero multiplication, then G is called a nil a -group. It is proved that a reduced Abelian torsionfree group G of finite rank is a T I-group if and only if G is a homogeneous Murley group or G is a nil a -group. We also study the interrelations between the class of homogeneous Murley groups and the class of nil a -groups. For any type t = (∞, ∞, . . .) и every integer n > 1, there exist 2 ℵ 0 pairwise nonquasi-isomorphic homogeneous Murley groups of type t and rank n which are nil a -groups. We describe types t such that there exists a homogeneous Murley group of type t which is not a nil a -group. This paper will be published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.

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“…In [AW04] the authors observed that the dualities * :A D : * preserve the torsion-free rank and the quasi-exactness in A and D (see also [Fo09]). Using Theorem 3.4 and the proof of Theorem 3.7, we have the following:…”

Section: Lemma 33 With the Notation Abovementioning

confidence: 99%

“…These numerous dualities were applied both to determine properties of the categories, such as the Krull-Schmidt property, and to establish existence of arbitrarily large indecomposable objects. Recently, Fomin proved in [Fo09] that the duality from [FoW981] comes from a duality between some subcategories of Ab, the category of all abelian groups, which have as objects finite rank torsion-free groups and quotient divisible groups respectively. Moreover this duality is exact.…”

Section: Introductionmentioning

confidence: 99%

Dualities for self–small groups

Breaz

Schultz

2011

Proc. Amer. Math. Soc.

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We construct a family of dualities on some subcategories of the quasi-category S of self-small groups of finite torsion-free rank which cover the class S. These dualities extend several of those in the literature. As an application, we show that a group A ∈ S is determined up to quasi-isomorphism by the Q-algebras {Q Hom(C, A) : C ∈ S} and {Q Hom(A, C) : C ∈ S}. We also generalize Butler's Theorem to self-small mixed groups of finite torsion-free rank.

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Invariants for abelian groups and dual exact sequences

Cited by 15 publications

References 18 publications

On a theorem of Stelzer for some classes of mixed groups

On a theorem of Stelzer for some classes of mixed groups

Rings on Abelian Torsion-Free Groups of Finite Rank

Dualities for self–small groups

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