Uncountable locally free groups and their group rings

TL;DR

This paper proves that uncountable locally free groups contain large free subgroups and establishes the primitivity of their group rings, extending previous results to uncountable cases.

Contribution

It demonstrates that uncountable locally free groups have large free subgroups and proves the primitivity of their group rings, generalizing prior work.

Findings

Uncountable locally free groups contain free subgroups of the same cardinality.

Group rings of locally free groups are primitive.

Extension of primitivity results to uncountable locally free groups.

Abstract

In this note, we show that an uncountable locally free group, and therefore every locally free group, has a free subgroup whose cardinality is the same as that of $G$. This result directly improve the main result in [T. Nishinaka,"Group rings of countable non-abelian locally free groups are primitive", Int. J. algebra and computation, 21(3)(2011), 409-431] and establish the primitivity of group rings of locally free groups.