ON ALGEBRAS ARISING FROM THE ELEMENTS OF A GALOIS GROUP FOR A GALOIS ALGEBRA

Let B be a ring with 1 and C the center of B. It is shown that if B is a Galois algebra over R with a finite Galois group G, Jg = {b ∈ B |bx = g(x)b for all x ∈ B} for each g ∈ G, and eg an idempotent in C such that BJg = Beg, then the algebra B(g) generated by {Jh |h ∈ G and eh = eg} for an g ∈ G is a separable algebra over Reg and a central weakly Galois algebra with Galois group K(g) generated by {h ∈ G |eh = eg}. Moreover, {B(g) |g ∈ G} and {K(g) |g ∈ G} are in a one-to-one correspondence, and three characterizations of a Galois extension are also given.