On separability problem for circulant S-rings

TL;DR

This paper establishes a criterion for separability of Schur rings over cyclic groups, proving all such rings over cyclic p-groups are separable and that the class of separable circulant S-rings is closed under duality.

Contribution

It introduces a new criterion for separability of S-rings over cyclic groups and proves all S-rings over cyclic p-groups are separable, also showing closure under duality.

Findings

All S-rings over cyclic p-groups are separable.

Separable circulant S-rings are closed under duality.

A criterion for separability of S-rings over cyclic groups is established.

Abstract

A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we prove that any S-ring over a cyclic $p$-group is separable and that the class of separable circulant S-rings is closed with respect to duality.