On ω-categorical groups and rings with NIP

2010/07/04 by Krzysztof Krupiński, Krzysztof Krupinski, Krupinski, Krzysztof
Computer Science · Mathematics · #03C35 #03C45 #20A15 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras #math.LO #msc:03C35 #msc:03C45 #msc:20A15

paper · pdf · doi:10.48550/arxiv.1007.0534

arxiv created 2010/07/04 · openalex publication_date 2010/07/04 · arxiv updated 2010/07/06 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the existence of a non-algebraic, generically stable over ∅ type, such a group is abelian.

Citations

• Superrosy dependent groups having finitely satisfiable generics
• On relationships between algebraic properties of groups and rings in some model-theoretic contexts

Related

• On omega-categorical groups and rings of finite burden
• Omega-categorical groups and rings of finite dimension
• Some remarks on dp-minimal groups
• Omega-categorical pseudofinite groups
• Definably amenable NIP groups