On Galois algebras satisfying the fundamental theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B-H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re circle plus R(1 - e) where e and I - e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V-B (A) = circle plus Sigma(g is an element of G(A)) J(g), and the centers of A and B-G(A) are the same where V-B(A) is the commutator subring of A in B, J(g) = {b is an element of B / bx = g(x)b for each X is an element of B) for a g is an element of G, and G(A) = {g is an element of G / g(a) = a for all a is an element of A}.