Mathieu-Zhao spaces over field of positive characteristic

Abstract

Let $K$ be a field of characteristic $p$, $\delta$ a nonzero $\mathcal{E}$-derivation and $D=f(x_1){\partial }_1$. We first prove that $\mathrm{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if $f(x_1)=x_1^r{f}_1(x_1^p)$ and $r\ne 1$. Then we prove that $\mathrm{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if $\delta$ 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 $K[x_1]$.

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