A GENERALIZATION OF THE DEMEYER THEOREM FOR CENTRAL GALOIS ALGEBRAS

Abstract

Let A be an Azumaya algebra over a semi-local ring R with no idempotents but 0 and 1, and M and N indecomposable nitely generated projective left A-modules. Then it was shown that M = N ([3], Theorem 1). Thus the Noether-Skolem theorem can be generalized from central simple algebras to Azumaya algebras over a semi-local ring with no idempotents but 0 and 1, that is, any automorphism of A is inner ([1], page 122). Consequently, any central Galois algebra over a semi-local ring with no idempotents but 0 and 1 is a projective group algebra ([1], Theorem 6). The purpose of the present paper is to generalize the above result to an Azumaya algebra A over a semi-local ring R (not necessarily with no idempotents but 0 and 1). LetM andN be nitely generated projective left A-modules. If the rank functions of M and N over R are equal, then M = N , where rankM (p) = the rank of the free Rp-module Mp over the local ring Rp at the prime ideal p of R. Then we shall show that the Noether-Skolem theorem holds for A, and a central Galois algebra over R with Galois group G is a projective group algebra of G over R, RGf , with a factor set f : G G ! funits of Rg as de ned by F. R. DeMeyer in [1]. Thus a Galois algebra (not necessarily central) over R can be shown to be a direct sum of projective group algebras.