## On Galois Extensions for Azumaya Group Rings

**Speaker(s): ** Professor Larry Xue (Bradley University)

**Time: ** 00:00-00:00 October 29, 2010

**Venue: ** Room 1328， Peking University Ziyuan Building

Title: On Galois Extensions for Azumaya Group Rings

Speaker: Professor Larry Xue (Bradley University)

Time: 14:00-14:45, October 29, 2010

Venus: Room 1328， Peking University Ziyuan Building

Abstract：

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 $oplussum_{overline gin overline K}J_{overline g}$ where $J_{overline g}={ain RG, ig|,ax=overline g(x)a$ for each $xin RG}$; and

(3) ${gin G, ig|,g$ is a representative of $overline gin overline K}$ are linearly independent over $C$.