A Correspondence Theorem for Galois Extensions of Rings

Let B be an indecomposable Galois extension of BG with Galois group G such that BG is a separable CG-algebra where C is the center of B. It is shown that B satisfies the fundamental theorem if and only if for each separable extension A of BG in B, VB(A) = ⊕∑g∈G(A) Jg, and the centers of A and BG(A) are the same where VB(A) is the com-mutator subring of A in B, Jg = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}. More-over, a correspondence theorem between subgroups of G and separable subextensions of BG in B is given.