On $\omega$-categorical groups and rings with NIP

TL;DR

This paper investigates the structure of omega-categorical groups and rings with NIP, establishing conditions under which these structures are nilpotent-by-finite or abelian, and exploring properties of specific types within these models.

Contribution

It proves that omega-categorical rings with NIP are nilpotent-by-finite and characterizes omega-categorical groups with NIP and fsg as nilpotent-by-finite, also analyzing the impact of strongly regular types.

Findings

Omega-categorical rings with NIP are nilpotent-by-finite.

Omega-categorical groups with NIP and fsg are nilpotent-by-finite.

Groups with strongly regular types are abelian.

Abstract

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