Bulletin of the
Korean Mathematical Society BKMS

Let $K$ be a field of characteristic zero. We first show that images of the linear derivations and the linear $\mathcal{E}$-derivations of the polynomial algebra $K[x]=K[x_1,x_2,\ldots,x_n]$ are ideals if the products of any power of eigenvalues of the matrices according to the linear derivations and the linear $\mathcal{E}$-derivations are not unity. In addition, we prove that the images of $D$ and $\delta$ are Mathieu-Zhao spaces of the polynomial algebra $K[x]$ if $D=\sum_{i=1}^n(a_ix_i+b_i)\partial_i$ and $\delta=I-\phi$, $\phi(x_i)=\lambda_ix_i+\mu_i$ for $a_i,b_i,\lambda_i,\mu_i\in K$ for $1\leq i\leq n$. Finally, we prove that the image of an affine $\mathcal{E}$-derivation of the polynomial algebra $K[x_1,x_2]$ is a Mathieu-Zhao space of the polynomial algebra $K[x_1,x_2]$. Hence we give an affirmative answer to the LFED Conjecture for the affine $\mathcal{E}$-derivations of the polynomial algebra $K[x_1,x_2]$.

Keywords: LFED conjecture, locally finite $\mathcal{E}$-derivations, Mathieu-Zhao spaces

MSC numbers: Primary 14R10, 13N15, 13F20

Supported by: The second author is supported by the NSF of China (Grant No. 11871241; 11601146), the China Scholarship Council and the Construct Program of the Key Discipline in Hunan Province.