Vol. 32, No. 1, 1970

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
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.