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Algebra Logika, 2017, Volume 56, Number 5, Pages 593–612 (Mi al818)  

This article is cited in 4 scientific papers (total in 4 papers)

Divisible rigid groups. Algebraic closedness and elementary theory

N. S. Romanovskiiab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090 Russia

Abstract: A group $G$ is said to be rigid if it contains a normal series
$$ G=G_1>G_2>…>G_m>G_{m+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and, treated as right $\mathbb Z[G/G_i]$-modules, are torsion-free. A rigid group $G$ is divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by nonzero elements of the ring $\mathbb Z[G/G_i]$. Every rigid group is embedded in a divisible one. We prove two theorems.
THEOREM 1. The following three conditions for a group $G$ are equivalent: $G$ is algebraically closed in the class $\Sigma_m$ of all $m$-rigid groups; $G$ is existentially closed in the class $\Sigma_m$; $G$ is a divisible $m$-rigid group.
THEOREM 2. The elementary theory of a class of divisible $m$-rigid groups is complete.

Keywords: divisible rigid group, algebraic closedness, elementary theory.

DOI: https://doi.org/10.17377/alglog.2017.56.505

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English version:
Algebra and Logic, 2017, 56:5, 395–408

Bibliographic databases:

UDC: 512.5+510.6
Received: 20.09.2015

Citation: N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra Logika, 56:5 (2017), 593–612; Algebra and Logic, 56:5 (2017), 395–408

Citation in format AMSBIB
\Bibitem{Rom17}
\by N.~S.~Romanovskii
\paper Divisible rigid groups. Algebraic closedness and elementary theory
\jour Algebra Logika
\yr 2017
\vol 56
\issue 5
\pages 593--612
\mathnet{http://mi.mathnet.ru/al818}
\crossref{https://doi.org/10.17377/alglog.2017.56.505}
\transl
\jour Algebra and Logic
\yr 2017
\vol 56
\issue 5
\pages 395--408
\crossref{https://doi.org/10.1007/s10469-017-9461-6}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000416984900005}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85035793341}


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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. A. G. Myasnikov, N. S. Romanovskii, “Divisible rigid groups. II. Stability, saturation, and elementary submodels”, Algebra and Logic, 57:1 (2018), 29–38  mathnet  crossref  crossref  isi
    2. N. S. Romanovskii, “Generalized rigid groups: definitions, basic properties, and problems”, Siberian Math. J., 59:4 (2018), 705–709  mathnet  crossref  crossref  isi  elib
    3. N. S. Romanovskii, “Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination”, Algebra and Logic, 57:6 (2019), 478–489  mathnet  crossref  crossref  isi
    4. 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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