On the Uniqueness of the Coefficient Ring in a Group Ring

Abstract

Let R1 and R2 be commutative rings with identities, G a group and R1G and R2G the group ring of G over R1 and R2 respectively. The problem that motivates this work is to determine what relations exist between R1 and R2 if R1G and R2G are isomorphic. For example, is the coefficient ring R1 an invariant of R1G? This is not true in general as the following example shows. Let H be a group andIf R1 is a commutative ring with identity and R2 = R1H, thenbut R1 needn't be isomorphic to R2.Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.