主 题: On Galois Extensions for Azumaya Group Rings
报告人: Prof. Lany XUE (Bradley University)
时 间: 2010-10-29 14:00-14:45
地 点: 资源大厦1328
Let $R$ be a ring with 1, $G$ a group, and $RG$ a group ring with center $C$. Assume $RG$
is an Azumaya $C$-algebra. Then the inner automorphism group $\overline G$ of $RG$ induced
by the elements of $G$ is finite, and $RG$ is not a Galois extension of $(RG)^{\overline G}$ with
Galois group $\overline G$. For a proper subgroup $\overline K$ of $\overline G$ with an
invertible order, the following are equivalent:
(1) $RG$ is a Galois extension of $(RG)^{\overline K}$ with Galois group
$\overline K$;
(2) $RG$ is a projective right $(RG)^{\overline K}$-module and the
centralizer of $(RG)^{\overline K}$ is $\oplus\sum_{\overline g\in \overline K}J_{\overline g}$ where
$J_{\overline g}=\{a\in RG\,\big|\,ax=\overline g(x)a$ for each $x\in RG\}$; and
(3) $\{g\in G\,\big|\,g$ is a representative of $\overline g\in \overline K\}$ are linearly independent over $C$.
