On Schur Rings over Infinite Groups III

Abstract

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group $\mathbb{Z}\times {\mathbb{Z}}_p$, where $p$ is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over $\mathbb{Z}\times {\mathbb{Z}}_p$ are traditional. We also draw analogs between Schur rings over $\mathbb{Z}\times {\mathbb{Z}}_p$ and partitions of difference sets over ${\mathbb{Z}}_p$.