论文标题
组环的扭曲推导问题
The Twisted Derivation Problem for Group Rings
论文作者
论文摘要
我们研究$（σ，τ）$ - 一个组环$ rg $的派生，其中$ g $是一组，中心，有有限指数为$ g $，$ r $是一个带有$ 1 $的半iprime戒指，这样要么$ r $没有扭曲元素，也没有$ r $ $ $ p $ $ p $ $ $ $ $ p $ $ p $ $ rg $的内态固定$ rg $的中心。我们概括了Main Theorem $ 1.1 $的\ cite {chau-19}，并证明有一个环$ t \ supset r $，以至于$ \ mathcal {z}（z}（t）\ supset \ supset \ mathcal {z}（z}（r）$及其自然扩展，$ t $ t $ tg $ tg $ tg $ tg $ tg $ tg $ tg $ tg $ tg $ tg $ tg $ $ h^1（tg，{}_σtg_τ）= 0 $，其中$ {}_σtg_τ$是扭曲的$ tg-tg $ -bimodule。我们将上述结果的应用和主定理的应用$ 1.1 $ \ cit {chau-19}到有限组的积分组环，并将整体组环的扭曲派生连接到现场的其他重要问题，例如同构问题和Zassenhaus的猜想。我们还举了一个既有本地有限又是nilpotent的组$ g $的示例，因此，对于每个字段$ f $，都存在$ f $ -linear $σ$ - $ fg $的$ fg $，不是$σ$ -inner。
We study $(σ,τ)$-derivations of a group ring $RG$ where $G$ is a group with center having finite index in $G$ and $R$ is a semiprime ring with $1$ such that either $R$ has no torsion elements or that if $R$ has $p$-torsion elements, then $p$ does not divide the order of $G$ and let $σ,τ$ be $R$-linear endomorphisms of $RG$ fixing the center of $RG$ pointwise. We generalize Main Theorem $1.1$ of \cite{Chau-19} and prove that there is a ring $T\supset R$ such that $\mathcal{Z}(T)\supset\mathcal{Z}(R)$ and that for the natural extensions of $σ, τ$ to $TG$ we get $H^1(TG,{}_σTG_τ)=0$, where ${}_σTG_τ$ is the twisted $TG-TG$-bimodule. We provide applications of the above result and Main Theorem $1.1$ of \cite{Chau-19} to integral group rings of finite groups and connect twisted derivations of integral group rings to other important problems in the field such as the Isomorphism Problem and the Zassenhaus Conjectures. We also give an example of a group $G$ which is both locally finite and nilpotent and such that for every field $F$, there exists an $F$-linear $σ$-derivation of $FG$ which is not $σ$-inner.
