On Schur Rings Over Infinite Groups

TL;DR

This paper extends the classification of Schur rings from finite cyclic groups to infinite torsion-free locally cyclic groups, broadening the understanding of their structure in infinite group contexts.

Contribution

It generalizes the Leung-Man classification of Schur rings from finite cyclic groups to all torsion-free locally cyclic groups, a significant extension in the theory.

Findings

Classification of Schur rings over finite cyclic groups

Extension of classification to torsion-free locally cyclic groups

Broader understanding of Schur rings in infinite groups

Abstract

Schur rings are a type of subring of the group ring that is spanned by a partition of the group that meets certain conditions. Past literature has exclusively focused on the finite group case. This paper extends many classic results about Schur rings to the infinite groups, including Leung-Man's classification of Schur rings over finite cyclic groups. Additionally, this classification will be extended to all torsion-free locally cyclic groups.