Mathieu-Zhao spaces over field of positive characteristic

TL;DR

This paper investigates Mathieu-Zhao spaces over fields of positive characteristic, characterizing when the images of certain derivations are Mathieu-Zhao spaces and classifying nilpotent derivations in this context.

Contribution

It provides a complete characterization of when the image of a derivation is a Mathieu-Zhao space over fields of positive characteristic, including classifications of nilpotent derivations.

Findings

Im D is not a Mathieu-Zhao space iff f(x_1)=x_1^r f_1(x_1^p) with r≠1

Im δ is a Mathieu-Zhao space iff δ is not locally nilpotent

Classification of some nilpotent derivations of K[x_1]

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 $\operatorname{Im}D$ 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 $\operatorname{Im}\delta$ 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 $K[x_1]$.