In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group \(\mathcal {Z}\times \mathcal {Z}_p\), 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 \(\mathcal {Z}\times \mathcal {Z}_p\) are traditional. We also draw analogs between Schur rings over \(\mathcal {Z}\times \mathcal {Z}_p\) and partitions of difference sets over \(\mathcal {Z}_p\).

在本文中，我们进一步发展了无限群上的 Schur 环的性质，特别强调虚拟循环群\(\mathcal {Z}\times \mathcal {Z}_p\)，其中p是素数。我们提供这些 Schur 环中本原集的结构定理。在费马和安全素数的情况下，证明了完整的分类定理，该定理表明\(\mathcal {Z}\times \mathcal {Z}_p\)上的所有 Schur 环都是传统的。我们还在\(\mathcal {Z}\times \mathcal {Z}_p\)上的 Schur 环和\(\mathcal {Z}_p\)上的差分集划分之间进行类比。