On weakly locally finite division rings

TL;DR

This paper constructs weakly locally finite division rings with arbitrary Gelfand-Kirillov dimensions, demonstrating their diversity beyond locally finite rings, and explores related algebraic questions such as the Kurosh Problem and Herstein's conjecture.

Contribution

It provides explicit constructions of weakly locally finite division rings with any Gelfand-Kirillov dimension, expanding understanding of their structure and properties.

Findings

Existence of weakly locally finite division rings with any Gelfand-Kirillov dimension

Demonstration that these rings are not necessarily locally finite

Investigation of algebraic questions related to the Kurosh Problem and Herstein's conjecture

Abstract

Weakly locally finite division rings were considered in \cite{dbh}, where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer $n\geq 0$ or $n=\infty$, we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is $n$. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well-known Kurosh Problem and with one of Herstein's conjectures.