Abstract
We prove that studying the universal theories of generalized rigid metabelian groups reduces to those of the pairs \( (A,R) \), where \( R \) is a commutative integral domain and \( A \) is a nontrivial torsion-free subgroup of the multiplicative group \( R^{\ast} \) generating \( R \).
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References
Romanovskii N. S., “Generalized rigid groups: Definitions, basic properties, and problems,” Sib. Math. J., vol. 59, no. 4, 705–709 (2018).
Romanovskii N. S., “Generalized rigid metabelian groups,” Sib. Math. J., vol. 60, no. 1, 148–152 (2019).
Romanovskii N. S., “Equational Noethericity of metabelian \( r \)-groups,” Sib. Math. J., vol. 61, no. 1, 154–158 (2020).
Chapuis O., “Universal theory of certain solvable groups and bounded Ore group rings,” J. Algebra, vol. 176, no. 2, 368–391 (1995).
Chapuis O., “\( \forall \)-free metabelian groups,” J. Symb. Log., vol. 62, no. 1, 159–174 (1997).
Romanovskii N. S., “Universal theories for free solvable groups,” Algebra and Logic, vol. 51, no. 3, 259–263 (2012).
Baumslag G., Myasnikov A., and Remeslennikov V., “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, vol. 219, no. 1, 16–79 (1999).
Romanovskii N. S., “Decomposition of a group over an Abelian normal subgroup,” Algebra and Logic, vol. 55, no. 4, 315–326 (2016).
Funding
The author was partially supported by the Mathematical Center in Akademgorodok under Agreement No. 075–15–2019–1613 with the Ministry of Science and Higher Education of the Russian Federation.
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Romanovskii, N.S. On the Universal Theories of Generalized Rigid Metabelian Groups. Sib Math J 61, 878–883 (2020). https://doi.org/10.1134/S0037446620050110
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DOI: https://doi.org/10.1134/S0037446620050110