Let G be a finite group of
automorphisms of a ring Λ which has identity element. Let C be the center of Λ, let
Γ be the subring of G-invariant elements of Λ, and assume that C is a separable
extension of C ∩ Γ. In the first section of this paper, it is shown that every finite
group of automorphisms of Λ over Γ is faithfully represented as a group of
automorphisms of C by restriction if, and only if, Λ = C ⊗C∩ΓΓ. Moreover, suppose
that Λ = C ⊗C∩ΓΓ and Ω is a subring of Λ such that Γ ⊆ Ω ⊆ Λ. Then there exists a
finite group H of automorphisms of Λ such that Ω is the subring of H-invariant
elements of Λ if, and only if, C ∩ Ω is a separable extension of C ∩ Γ and
Ω = (C ∩ Ω) ⊗C∩ΓΓ.
Let R be a commutative ring with identity element; and assume now that Λ is a
separable algebra over R and G is a finite group of automorphisms of the
R-algebra Λ. In the second section of this paper, it is shown that C is the
centralizer of Γ in Λ if, and only if, Λ = C ⊗C∩ΓΓ. Moreover, suppose that
Λ = C ⊗C∩ΓΓ and Ω is a subalgebra of Λ such that Γ ⊆ Ω ⊆ Λ. Then there
exists a finite group H of automorphisms of Λ such that Ω is the subalgebra
of H-invariant elements of Λ if, and only if, Ω is a separable algebra over
R.
