Simple derivations and their images

Abstract

In the paper, we prove that the derivation $D=y{\partial }_x+(a_2(x)y^2+a_1(x)y+a_0(x)){\partial }_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^{\ast }$, $(2)$ $°a_1(x)\ge 1$ or $°a_2(x)\ge 1$, $(3)$ there exist no $l\in K^{\ast }$ such that $a_2(x)=l{a}_1(x)-l^2{a}_0(x)$. In addition, we prove that the image of the derivation $D={\partial }_x+{\sum }_{i=1}^n{\gamma }_i(x)y_i^{k_i}{\partial }_i$ is a Mathieu-Zhao space iff $D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum }_{i=1}^n{\gamma }_i{y}_i^{k_i}{\partial }_i$ of $K[y_1,\dots ,y_n]$ is a Mathieu-Zhao space iff $k_i\le 1$ for all $1\le i\le n$, $n\ge 2$.

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