Schur rings over a product of Galois rings

Abstract

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring \(R\) being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of \(R\) is odd, then as in the cyclic group case any pure Schur ring over \(R\) is the tensor product of a pure cyclotomic ring and Schur rings of rank \(2\) over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over \(R\) that are not generalized wreath products.