IJPAM: Volume 31, No. 3 (2006)


Ted Hurley
Department of Mathematics
National University of Ireland
e-mail: ted.hurley@nuigalway.ie

Abstract.It is shown that the group ring $RG$ of a group $G$ of order $n$ over a ring $R$ is isomorphic to a certain ring of $n\times n$ matrices over $R$. When the ring $R$ has an identity element and no zero-divisors, this representation enables us to describe the units and zero-divisors of the group ring in terms of properties of these matrices and where appropriate in terms of the determinant of the matrices.

The isomorphism extends to group rings of infinite groups when the elements of the group can be listed.

The rings of matrices which turn up as isomorphic to certain group rings include circulant matrices, Toeplitz matrices, Walsh-Toeplitz matrices, circulant or Toeplitz combined with Hankel matrices and block-type circulant matrices. Group rings thus can be considered to be a generalisation of these rings of matrices, which occur in communications, signal processes, time series analysis and elsewhere.

When $G$ is finite and $R$ is a field, it follows from the representation that $U(RG)$, the group of units of $RG$, satisfies the Tits' alternative and consequently the generalised Burnside problem has a positive answer for $U(RG)$.

Received: July 19, 2006

AMS Subject Classification: 20C05, 20C07, 16S34, 15A30

Key Words and Phrases: group ring, rings of matrices

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 31
Issue: 3