Notes on Galois Extensions with Inner Galois Groups

Xiao-Long Jiang and George Szeto

Abstract: Let $S$ be a ring with 1, $C$ the center of $S$, $G$ a finite inner automorphism group of $S$ of order $n$ for some integer $n$ invertible in $S$ where $G=\{g_{1},g_{2},...,g_{n}\}$ and $g_{i}(s)=U_{i}\,s\,U_{i}^{-1}$ for some $U_{i}$ in $S$ and all $s$ in $S$, and $R$ the subring of all elements fixed under each element in $G$. Then, $S$ is a $G$-Galois extension of $R$ which is an Azumaya $C$-algebra with a Galois system $\{n^{-1}U_{i},\,U_{i}^{-1}\}$ if and only if $S$ is a projective group ring $RG_{f}$ for some factor set $f$ which is an $H$-separable extension of $R$ and $R$ is a separable $C$-algebra. Moreover, some correspondence relations are given between certain sets of separable subalgebras of such an $S$.

Keywords: Galois extensions; projective group rings; Azumaya algebras; $H$-separable extensions.

Classification (MSC2000): 16S30, 16W20.

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