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: $(1)$ $a_0(x)\in K^*$, $(2)$ $\deg a_1(x)\geq1$ or $\deg a_2(x)\geq1$, $(3)$ there exist no $l\in K^*$ such that $a_2(x)=la_1(x)-l^2a_0(x)$. 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 $K[y_1,\ldots,y_n]$ is a Mathieu-Zhao space iff $k_i\leq 1$ for all $1\leq i\leq n$, $n\geq 2$.