Schur rings over a product of Galois rings

TL;DR

This paper generalizes the theory of Schur rings from cyclic groups to rings formed by products of Galois rings, revealing new structural properties and examples of non-pure Schur rings.

Contribution

It extends Schur ring theory to Galois ring products, characterizes pure Schur rings, and identifies non-pure Schur rings not arising from wreath products.

Findings

Pure Schur rings over R are tensor products of cyclotomic rings and rank 2 Schur rings.

If the characteristic of R is odd, pure Schur rings have a specific structure.

Existence of non-pure Schur rings over R that are not generalized wreath products.

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.