• Dr. Hiteshwar Singh
• Dhananjay Kumar Mishra

It is shown that the group ring R[G] of a group G of order n over a ring R is isomorphic to a certain ring of n × 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 as well as 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 is 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 R[G] thus can be considered to be a generalization of these rings of matrices. These generalizations are used 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(R[G]), the group of units of R[G], satisfies the Tits’ alternative and consequently the generalized Burnside problem has a positive answer for U(R[G]).

group ring, rings of matrices, circulant matrices, Toeptitize matrices, Walsh-Toeplitz matrices, Hankel matrices, Tits alternative

"RELATION BETWEEN GROUP RINGS R[G] AND RINGS OF MATRICES M(R[G], w), w R[G]", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.11, Issue 11, page no.g305-g310, November-2024, Available :http://www.jetir.org/papers/JETIR2411627.pdf

"RELATION BETWEEN GROUP RINGS R[G] AND RINGS OF MATRICES M(R[G], w), w R[G]", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.11, Issue 11, page no. ppg305-g310, November-2024, Available at : http://www.jetir.org/papers/JETIR2411627.pdf