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Algebra Logika, 2023 Volume 62, Number 1, Pages 102–113 (Mi al2750)

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.

UDC: 512.5:510.6

Received: 22.02.2022
Revised: 30.10.2023

DOI: 10.33048/alglog.2023.62.107



© Steklov Math. Inst. of RAS, 2024