A matrix representation of an Azumaya group ring

Abstract

Let R be an indecomposable ring with 1 of characteristic p ^k for some prime integer p and integer k, G a group and RG the group ring of G over R. It is shown that if RG is an Azumaya algebra, then RG contains a direct sum of matrix rings over Azumaya algebras, and RG is a direct sum of matrix rings over Azumaya algebras if and only if the centre of RCG′ is C where C is the centre of RG, G′ is the commutator subgroup of G, and RCG′ = {abc|a ∈ R, b ∈ C, c ∈ G′}. Moreover, when the characteristic of R is 0, an expression of an Azumaya group ring is also obtained.

Keywords:

• group algebras
• group rings
• Azumaya algebras

AMS Subject Classifications::

• 16S35
• 16D70

Acknowledgements

This article was revised according to the suggestions of the referee and was supported by a Caterpillar Fellowship at Bradley University. The authors thank the referee for the valuable suggestions and Caterpillar Inc. for the support.