Vol. 32, No. 1, 1970

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Outer Galois theory for separable algebras

Herbert Frederick Kreimer, Jr.

Vol. 32 (1970), No. 1, 147–155
Abstract

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 CΓΓ. Moreover, suppose that Λ = C CΓΓ 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) CΓΓ.

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 CΓΓ. Moreover, suppose that Λ = C CΓΓ 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.

Mathematical Subject Classification
Primary: 16.70
Milestones
Received: 18 February 1969
Published: 1 January 1970
Authors
Herbert Frederick Kreimer, Jr.