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Notes on Galois extensions with inner Galois groups. (English) Zbl 0939.16017

Let \(S\) be a ring with 1 and let \(G\) be a finite subgroup of inner automorphisms of \(S\) with \(|G|^{-1}\in S\), and \(g_i(s)=U_isU_i^{-1}\), for each \(g_i\in G\) and \(U_i\) in \(S\). This paper gives a slight modification of a result by F. R. DeMeyer [Ill. J. Math. 10, 287-295 (1966; Zbl 0216.34001)], changing the hypothesis of the fixed ring \(R\) being the center of \(S\) to specifying the canonical Galois system \(\{|G|^{-1}U_i,U_i^{-1}\}\). With this revised condition, the authors then show that the following three conditions are equivalent: \(S\) is an Azumaya algebra, \(S\) is an \(H\)-separable extension of \(R\) and \(R\) is separable over \(C\), and \(R\) is an Azumaya \(C\)-algebra. Furthermore, the authors give one-to-one correspondences between the separable \(C\)-subalgebras of \(S\) containing (contained in) \(R\) with the \(C\)-subalgebras of \(S\) contained in (containing) \(CG_f\).
Reviewer: R.Alfaro (Flint)

MSC:

16S35 Twisted and skew group rings, crossed products
16W20 Automorphisms and endomorphisms
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)

Citations:

Zbl 0216.34001
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