Dimensional groups and fields

TL;DR

This paper introduces a general notion of dimension in algebraic structures and explores its implications for groups and rings, including chain conditions, definability of fields, and properties of pseudofinite groups.

Contribution

It develops a unified framework for dimension in algebraic structures and derives new structural results for groups and rings under this notion.

Findings

Pseudofinite groups contain large finite-by-abelian subgroups.

Pseudofinite groups of dimension 2 contain large soluble subgroups.

Chain conditions are established for groups based on the new dimension concept.

Abstract

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that pseudofinite groups contain big finite-by-abelian subgroups, and pseudofinite groups of dimension 2 contain big soluble subgroups.