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Algebra Logika, 2009, Volume 48, Number 2, Pages 258–279 (Mi al399)  

This article is cited in 24 scientific papers (total in 24 papers)

Equational Noetherianness of rigid soluble groups

N. S. Romanovskii

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Abstract: A group $G$ is said to be rigid if it contains a normal series of the form
$$ G=G_1>G_2>…>G_m>G_{m+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and are torsion free as right $Z[G/G_i]$-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any $n$, every system of equations in $x_1,…,x_n$ over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on $G^n$, which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian.

Keywords: rigid group, equational Noetherianness.

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English version:
Algebra and Logic, 2009, 48:2, 147–160

Bibliographic databases:

UDC: 512.5
Received: 05.09.2008

Citation: N. S. Romanovskii, “Equational Noetherianness of rigid soluble groups”, Algebra Logika, 48:2 (2009), 258–279; Algebra and Logic, 48:2 (2009), 147–160

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. S. Romanovskii, “Irreducible algebraic sets over divisible decomposed rigid groups”, Algebra and Logic, 48:6 (2009), 449–464  mathnet  crossref  mathscinet  zmath  isi
    2. N. S. Romanovskii, “Coproducts of rigid groups”, Algebra and Logic, 49:6 (2010), 539–550  mathnet  crossref  mathscinet  isi
    3. A. G. Myasnikov, N. S. Romanovskii, “Universal theories for rigid soluble groups”, Algebra and Logic, 50:6 (2012), 539–552  mathnet  crossref  mathscinet  zmath  isi
    4. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. II. Foundations”, J. Math. Sci., 185:3 (2012), 389–416  mathnet  crossref
    5. Romanovskiy N.S., “Presentations for Rigid Solvable Groups”, J. Group Theory, 15:6 (2012), 793–810  crossref  mathscinet  zmath  isi  elib  scopus
    6. M. V. Kotov, “Topologizability of countable equationally Noetherian algebras”, Algebra and Logic, 52:2 (2013), 105–115  mathnet  crossref  mathscinet  isi
    7. N. S. Romanovskii, “Irreducibility of an affine space in algebraic geometry over a group”, Algebra and Logic, 52:3 (2013), 262–265  mathnet  crossref  mathscinet  isi
    8. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113  mathnet  crossref  mathscinet  isi
    9. D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra and Logic, 53:2 (2014), 133–139  mathnet  crossref  mathscinet  isi
    10. Myasnikov A.G. Romanovskii N.S., “Logical Aspects of the Theory of Divisible Rigid Groups”, Dokl. Math., 90:3 (2014), 697–698  crossref  mathscinet  zmath  isi  elib  scopus
    11. E. I. Timoshenko, “Normal automorphisms of a soluble product of abelian groups”, Siberian Math. J., 56:1 (2015), 191–198  mathnet  crossref  mathscinet  isi  elib  elib
    12. N. S. Romanovskiy, “Hilbert's Nullstellensatz in algebraic geometry over rigid soluble groups”, Izv. Math., 79:5 (2015), 1051–1063  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. Ch. K. Gupta, N. S. Romanovskii, “$\mathbb Q$-completions of free solvable groups”, Algebra and Logic, 54:2 (2015), 127–139  mathnet  crossref  crossref  mathscinet  isi
    14. N. S. Romanovskii, “Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group”, Algebra and Logic, 54:6 (2016), 478–488  mathnet  crossref  crossref  mathscinet  isi
    15. N. S. Romanovskii, “Decomposition of a group over an Abelian normal subgroup”, Algebra and Logic, 55:4 (2016), 315–326  mathnet  crossref  crossref  isi
    16. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386  mathnet  crossref  crossref  isi
    17. A. G. Myasnikov, N. S. Romanovskii, “Model-theoretic aspects of the theory of divisible rigid soluble groups”, Algebra and Logic, 56:1 (2017), 82–84  mathnet  crossref  crossref  isi
    18. V. A. Roman'kov, “Solvability of equations in classes of solvable groups and Lie algebras”, Algebra and Logic, 56:3 (2017), 251–255  mathnet  crossref  crossref  mathscinet  isi
    19. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometric equivalence”, Algebra and Logic, 56:4 (2017), 281–294  mathnet  crossref  crossref  isi
    20. N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408  mathnet  crossref  crossref  isi
    21. S. G. Afanas'eva, “Algebraic sets in a divisible $2$-rigid group”, Siberian Math. J., 59:2 (2018), 202–206  mathnet  crossref  crossref  isi  elib
    22. N. S. Romanovskii, “Generalized rigid groups: definitions, basic properties, and problems”, Siberian Math. J., 59:4 (2018), 705–709  mathnet  crossref  crossref  isi  elib
    23. Myasnikov A.G., Romanovskii N.S., “Characterization of Finitely Generated Groups By Types”, Int. J. Algebr. Comput., 28:8, SI (2018), 1613–1632  crossref  mathscinet  zmath  isi  scopus
    24. N. S. Romanovskii, “Generalized rigid metabelian groups”, Siberian Math. J., 60:1 (2019), 148–152  mathnet  crossref  crossref  isi
  • Алгебра и логика Algebra and Logic
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