简单推导及其图像
Simple derivations and their images
摘要：
本文证明了用$a_2(x),a_1(x),a_0(x)\in K[x]$推导$K[x,y]$的$D=y\partial_x+(a_2(x)y^2+a_1(x)y+a_0(x))\partial_y$是简单的当且仅当下列条件成立：$(1)$ $a_0(x)\in K^*$，$(2)$ $°a_1(x)\geq1$或$°a_2(x)\geq1$，$(3)$不存在$l\in K^*$使得$a_2(x)=la_1(x)-l^2a_0(x)$ .另外，我们证明了导子$D=\partial_x+{\sum_{i=1}^n γ_i(x) y_i^{k_i}}{\partial_i}$的象是Mathieu-Zhao空间当且仅当$D$是局部有限的.证明了$K[y_1,\ldots,y_n]$的导子$D={\sum_{i=1}^n γ_i y_i^{k_i}}{\partial_i}$的象是Mathieu-Zhao空间当且仅当对所有$1\leq i\leq n$，$n\geq 2$ .
英文摘要：
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: $(1)$ $a_0(x)\in K^*$, $(2)$ $°a_1(x)\geq1$ or $°a_2(x)\geq1$, $(3)$ 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 $D=\partial_x+{\sum_{i=1}^n γ_i(x) y_i^{k_i}}{\partial_i}$ is a Mathieu-Zhao space iff $D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum_{i=1}^n γ_i y_i^{k_i}}{\partial_i}$ of $K[y_1,\ldots,y_n]$ is a Mathieu-Zhao space iff $k_i\leq 1$ for all $1\leq i\leq n$, $n\geq 2$.
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