A characterization of algebraic structures of group rings obtained from matrix rings

Abstract

Let R be a commutative ring with unity and G a finite group. We investigate when a group ring R[G] is isomorphic to a full matrix ring M_n(S) for some (possibly non-commutative) ring S. For commutative R with |G| invertible, we prove that such an isomorphism forces G to be trivial when n > 1. Thus the only non-trivial matrix ring isomorphisms for group rings occur when |G| is not invertible in R or when R itself is non-commutative. We also give a complete description of embeddings R[G]↪Mn​(T) as direct summands in terms of normal subgroups, and we discuss consequences for Morita equivalence and maximal orders. Examples illustrate the optimality of our conditions, and we conclude with open problems.