Invariants of Group Rings

Since the foundation paper of M. Auslander and 0. Goldman [ 1 J, several authors have generalized constructions for studying group algebras over fields to separable algebras over commutative rings. Specifically, we have in mind the Schur index [20], the Schur exponent [18], the Schur group [S], and the uniform group [ 111. This paper gives additional properties of these constructions and studies the relationship between them. First we reproduce the definitions of the Schur index M,(R) and Schur exponent m,(R) of a character x on a finite group G with respect to a commutative ring R in which [G: 11 is a unit. If R is a held then m,(R) = M,(R) and more generally if R is a commutative ring then m,(R) 1 M,(R). We conjecture m,(R) = M,(R) for every commutative ring R. If Y(R) denotes the Schur group of R, %(R) the uniform group of R, and B(R) the Brauer group of R then when R is a field Y(R) c %(R) c B(R). Each of these inclusions may be proper. Here we are mostly interested in Q(R) and