Galois Extensions with a Galois Commutator Subring

Let B be a Galois extension of B G with Galois group G, Δ the commutator subring of B G in B, and G|Δ the restriction of G to Δ. Equivalent conditions are given for a Galois extension Δ of Δ G with Galois group G|Δ. It is shown that the following statements are equivalent: (1) Δ is a Galois extension of Δ G with Galois group induced by and isomorphic with G/N where N = {g ∈ G |g(x )= x for all x ∈ Δ}. (2) B G Δ is a Galois extension of B G with Galois group induced by and isomorphic with G/N and Δ is a finitely generated and projective module over Δ G . (3) B is a composition of two Galois extensions: B ⊃ B G Δ with Galois group N and B G Δ ⊃ B G with Galois group induced by and isomorphic with G/N such that Δ is a finitely generated and projective module over Δ G . Consequently, more results can be derived for several well known classes of Galois extensions such as DeMeyer-Kanzaki Galois extensions, Azumaya Galois extensions, and Hirata separable Galois extensions.