Group rings of countable non-abelian locally free groups are primitive

TL;DR

This paper proves that group rings of countable non-abelian locally free groups are primitive, extending known results and providing conditions for primitive group rings of certain HNN extensions using graph theory and Formanek's method.

Contribution

It establishes the primitivity of group rings for a broad class of non-abelian locally free groups and extends existing results to uncountable cases with new graph-theoretic techniques.

Findings

Group rings of countable non-abelian locally free groups are primitive.

Provides necessary and sufficient conditions for primitivity of group rings of HNN extensions.

Extends primitivity results from countable to uncountable cardinalities.

Abstract

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the known result for the countable case to the general cardinality case. In order to prove the main theorem, we state some graph-theoretic results and apply them to to Formanek's method.