On relationships between algebraic properties of groups and rings in some model-theoretic contexts

TL;DR

This paper explores the connections between algebraic properties of groups and rings within specific model-theoretic frameworks, leading to structural insights and relationships among conjectures in the context of $ ext{omega}$-categorical and compact $G$-rings.

Contribution

It establishes new relationships between algebraic properties of definable groups and rings in model theory, providing structural results for $ ext{omega}$-categorical and small, $nm$-stable compact $G$-rings.

Findings

Structural results for $ ext{omega}$-categorical rings

Relationships between conjectures on small profinite groups

Connections between algebraic properties in model-theoretic contexts

Abstract

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings as well as about small, $nm$-stable compact $G$-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.