Glasnik Matematicki, Vol. 37, No.1 (2002), 13-19.

NOTES ON GALOIS ALGEBRAS

George Szeto and Lianyong Xue

Abstract.   Let B be a ring with 1, C the center of B, and G an automorphism group of B of order n for some integer n. Assume B is a Galois algebra over R with Galois group G. For a nonzero idempotent e ∈ R, it the rank of Be over Ce is defined and equal to the order of H|_Be where H = {g ∈ G | g(c) = c for each c ∈ C}, then Be is a central Galois algebra with Galois group H|_Be. This generalizes the F. R. DeMeyer and T. Kanzaki theorems for Galois algebras. Moreover, a structure theorem for a Galois algebra is given in terms of the concept of the rank of projective module.

2000 Mathematics Subject Classification.   16S35, 16W20.

Key words and phrases.   Galois extensions, Galois algebras, central Galois extensions, separable extensions, Azumaya algebras, rank of a projective module.