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Algebra Logika, 2009, Volume 48, Number 6, Pages 793–818 (Mi al424)  

This article is cited in 20 scientific papers (total in 20 papers)

Irreducible algebraic sets over divisible decomposed rigid groups

N. S. Romanovskiiab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: A soluble group $G$ is said to be rigid if it contains a normal series of the form
$$ G=G_1>G_2>…>G_p>G_{p+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and are torsion-free when treated as right $\mathbb Z[G/G_i]$-modules. Free soluble groups are important examples of rigid groups. A rigid group $G$ is divisible if elements of a quotient $G_i/G_{i+1}$ are divisible by nonzero elements of a ring $\mathbb Z[G/G_i]$, or, in other words, $G_i/G_{i+1}$ is a vector space over a division ring $Q(G/G_i)$ of quotients of that ring. A rigid group $G$ is decomposed if it splits into a semidirect product $A_1A_2…A_p$ of Abelian groups $A_i\cong G_i/G_{i+1}$. A decomposed divisible rigid group is uniquely defined by cardinalities $\alpha_i$ of bases of suitable vector spaces $A_i$, and we denote it by $M(\alpha_1,…,\alpha_ p)$.
The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR]], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [Algebra i Logika, <b>48</b>:2 (2009), 258–279], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group $M(\alpha_1,…,\alpha_ p)$. Our present goal is to derive important information directly about algebraic geometry over $M(\alpha_1,…,\alpha_ p)$. Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over $M(\alpha_1,…,\alpha_ p)$ using the language of equations.

Keywords: algebraic geometry, irreducible algebraic set, rigid group, universally equivalent groups.

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English version:
Algebra and Logic, 2009, 48:6, 449–464

Bibliographic databases:

UDC: 512.542
Received: 15.08.2009

Citation: N. S. Romanovskii, “Irreducible algebraic sets over divisible decomposed rigid groups”, Algebra Logika, 48:6 (2009), 793–818; Algebra and Logic, 48:6 (2009), 449–464

Citation in format AMSBIB
\by N.~S.~Romanovskii
\paper Irreducible algebraic sets over divisible decomposed rigid groups
\jour Algebra Logika
\yr 2009
\vol 48
\issue 6
\pages 793--818
\jour Algebra and Logic
\yr 2009
\vol 48
\issue 6
\pages 449--464

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    This publication is cited in the following articles:
    1. N. S. Romanovskii, “Coproducts of rigid groups”, Algebra and Logic, 49:6 (2010), 539–550  mathnet  crossref  mathscinet  isi
    2. 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
    3. Romanovskiy N.S., “Presentations for Rigid Solvable Groups”, J. Group Theory, 15:6 (2012), 793–810  crossref  mathscinet  zmath  isi  elib  scopus
    4. 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
    5. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113  mathnet  crossref  mathscinet  isi
    6. D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra and Logic, 53:2 (2014), 133–139  mathnet  crossref  mathscinet  isi
    7. 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
    8. 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
    9. 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
    10. 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
    11. N. S. Romanovskii, “Decomposition of a group over an Abelian normal subgroup”, Algebra and Logic, 55:4 (2016), 315–326  mathnet  crossref  crossref  isi
    12. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386  mathnet  crossref  crossref  isi
    13. 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
    14. 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
    15. 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
    16. N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408  mathnet  crossref  crossref  isi
    17. 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
    18. N. S. Romanovskii, “Generalized rigid groups: definitions, basic properties, and problems”, Siberian Math. J., 59:4 (2018), 705–709  mathnet  crossref  crossref  isi  elib
    19. 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
    20. N. S. Romanovskii, “Generalized rigid metabelian groups”, Siberian Math. J., 60:1 (2019), 148–152  mathnet  crossref  crossref  isi
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