On Adjoint Entropy of Abelian Groups

Goldsmith, Brendan; Gong, Ketao

doi:10.1080/00927872.2010.543447

Cited by 16 publications

(13 citation statements)

References 13 publications

(19 reference statements)

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“…We thank Professor Dikranjan for his very useful comments and suggestions, and also Professor Salce and Professor Göbel for their suggestions about already existing examples and results, which helped us in completing this paper. We are also grateful to Professor Goldsmith for giving us preprints of [15] and [16], and to the referee, who gave us the possibility to improve the exposition of our results with many useful observations.…”

Section: Aknowledgementsmentioning

confidence: 99%

String Numbers of Abelian Groups

Bruno

Virili

2012

J. Algebra Appl.

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The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function. Later on its values for endomorphisms of abelian groups were calculated in full generality. We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups. Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties.

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Section: Aknowledgementsmentioning

confidence: 99%

String Numbers of Abelian Groups

Bruno

Virili

2012

J. Algebra Appl.

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“…Much research dedicated to algebraic entropy have been made in recent years; we refer to [3] for a comprehensive bibliography on this subject. The parallel (but not exactly "dual") notion of adjoint entropy was introduced and investigated by Dikranjan, Giordano Bruno, and Salce in [4], and studied in several other papers, noticeably [5]. See again [3] for the bibliography.…”

Section: Introductionmentioning

confidence: 99%

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Zanardo

2014

Communications in Algebra

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Let R be a commutative ring and A be an R-module. The Mal'cev rank A of A is the sup of genN , where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N . We prove that is both subadditive and pre-additive as an invariant of Mod R . Our main goal is to investigate for modules over pseudo-valuation domains. Specifically, we establish which pseudovaluation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class of pseudo-valuation domains as a union = 1 ∪ 2 ∪ 3 ∪ 4 of suitably defined subclasses, and prove that the property holds if and only if R ∈ 3 ∪ 4 . In that case we can describe the R-modules A where A < . We also show that, for R ∈ 4 , there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.

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“…Abelian Hopfian and co-Hopfian groups have arisen recently in the study of algebraic entropy and its dual, adjoint entropy -see e.g. [4,7,9]. Despite the seeming simplicity of their definitions, Hopfian and co-Hopfian groups are notoriously difficult to handle and easily stated problems have remained open for a long time: if G is Hopfian, is the direct product G × Z Hopfian?…”

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confidence: 99%