The Invariant Subrings of Demeyer--Kanzaki Galois Extensions

Lianyong Xue

Abstract: Let $B$ be a ring with 1, $G$ a finite automorphism group of $B$, $C$ the center of $B$, $B^{G}$ the set of elements in $B$ fixed under each element in $G$. When $B$ is a DeMeyer--Kanzaki Galois extension of $B^G$ with Galois group $G$, it was shown that a separable subring $S$ of $B$ over $B^G$ is equal to $B^K$ for some subgroup $K$ of $G$ if and only if $CJ_g^{(S)}$ is a faithful $C$-module for each $g\not\in K$ where $J_g^{(S)}=\{s-g(s)\,|\; s\in S\}$. Moreover, the invariant subrings of $C$ over $C^G$ (i.e., $S=C^K$ for some subgroup $K$ of $G$) and of $B\ast G$ over $(B\ast G)^{\bar G}$ are characterized in terms of the faithful $B$-module $BJ_g^{(S)}$ and the faithful $C^G$-module ${C^G}J_g^{(S)}$ respectively for $g\in G$.

Keywords: Galois extensions; center Galois extensions; DeMeyer--Kanzaki Galois extensions; separable extensions; $H$-separable extensions; Azumaya algebras.

Classification (MSC2000): 16S35; 16W20.

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