论文标题
Mathieu-Zhao在积极特征领域的空间
Mathieu-Zhao spaces over field of positive characteristic
论文作者
论文摘要
令$ k $为特征$ p $，$δ$ a nonzero $ \ mathcal {e} $ - 推导和$ d = f（x_1）\ partial_1 $。我们首先证明$ \ operatorName {im} d $不是$ k [x_1] $的MATHIEU-ZHAO空间，并且仅当$ f（x_1）= x_1 = x_1^rf_1（x_1^p）$和$ r \ r \ r \ neq 1 $。然后，我们证明$ \ operatorname {im}δ$是$ k [x_1] $的MATHIEU-ZHAO空间，并且仅当$δ$不是本地nilpotent。最后，我们将一些$ K [x_1] $的nilpotent推导分类，并为$ d（i）$提供足够且必要的条件，使其成为$ k [x_1] $的Mathieu-zhao空间，用于任何理想的$ i $ a $ a $ k [x_1] $。
Let $K$ be a field of characteristic $p$, $δ$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. We first prove that $\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. Then we prove that $\operatorname{Im}δ$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $δ$ is not locally nilpotent. Finally, we classify some nilpotent derivations of $K[x_1]$ and give a sufficient and necessary condition for $D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$.
