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Generic types and generic elements in divisible rigid groups
A. G. Myasnikova, N. S. Romanovskiib a Charles V. Schaefer, Jr. School of Engineering & Science, Stevens Institute of Technology
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Abstract:
A group $G$ is said to be $m$-rigid if it contains a normal series of the form $$G=G_1>G_2>\ldots>G_m>G_{m+1}=1,$$ 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 said to be divisible if elements of the quotient $\rho_i(G)/\rho_{i+1}(G)$ are divisible by nonzero elements of the ring ${\mathbb{Z}}[G/\rho_i(G)]$. Previously, it was proved that the theory of divisible $m$-rigid groups is complete and $\omega$-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible $m$-rigid group $G$.
Keywords:
divisible $m$-rigid group, generic type, generic element.
Received: 22.02.2022 Revised: 30.10.2023
Citation:
A. G. Myasnikov, N. S. Romanovskii, “Generic types and generic elements in divisible rigid groups”, Algebra Logika, 62:1 (2023), 102–113
Linking options:
https://www.mathnet.ru/eng/al2750 https://www.mathnet.ru/eng/al/v62/i1/p102
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Abstract page: | 54 | Full-text PDF : | 14 | References: | 7 |
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