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 Algebra Logika: Year: Volume: Issue: Page: Find

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

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

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
\Bibitem{Rom09} \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 \mathnet{http://mi.mathnet.ru/al424} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2640965} \zmath{https://zbmath.org/?q=an:1245.20054} \transl \jour Algebra and Logic \yr 2009 \vol 48 \issue 6 \pages 449--464 \crossref{https://doi.org/10.1007/s10469-009-9071-z} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000273168500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77949270192} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. S. Romanovskii, “Coproducts of rigid groups”, Algebra and Logic, 49:6 (2010), 539–550
2. A. G. Myasnikov, N. S. Romanovskii, “Universal theories for rigid soluble groups”, Algebra and Logic, 50:6 (2012), 539–552
3. Romanovskiy N.S., “Presentations for Rigid Solvable Groups”, J. Group Theory, 15:6 (2012), 793–810
4. N. S. Romanovskii, “Irreducibility of an affine space in algebraic geometry over a group”, Algebra and Logic, 52:3 (2013), 262–265
5. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113
6. D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra and Logic, 53:2 (2014), 133–139
7. Myasnikov A.G. Romanovskii N.S., “Logical Aspects of the Theory of Divisible Rigid Groups”, Dokl. Math., 90:3 (2014), 697–698
8. N. S. Romanovskiy, “Hilbert's Nullstellensatz in algebraic geometry over rigid soluble groups”, Izv. Math., 79:5 (2015), 1051–1063
9. Ch. K. Gupta, N. S. Romanovskii, “$\mathbb Q$-completions of free solvable groups”, Algebra and Logic, 54:2 (2015), 127–139
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
11. N. S. Romanovskii, “Decomposition of a group over an Abelian normal subgroup”, Algebra and Logic, 55:4 (2016), 315–326
12. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386
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
14. V. A. Roman'kov, “Solvability of equations in classes of solvable groups and Lie algebras”, Algebra and Logic, 56:3 (2017), 251–255
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
16. N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408
17. S. G. Afanas'eva, “Algebraic sets in a divisible $2$-rigid group”, Siberian Math. J., 59:2 (2018), 202–206
18. N. S. Romanovskii, “Generalized rigid groups: definitions, basic properties, and problems”, Siberian Math. J., 59:4 (2018), 705–709
19. Myasnikov A.G., Romanovskii N.S., “Characterization of Finitely Generated Groups By Types”, Int. J. Algebr. Comput., 28:8, SI (2018), 1613–1632
20. N. S. Romanovskii, “Generalized rigid metabelian groups”, Siberian Math. J., 60:1 (2019), 148–152
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