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http://dx.doi.org/10.4134/BKMS.b210035

THE IMAGES OF LOCALLY FINITE 𝓔-DERIVATIONS OF POLYNOMIAL ALGEBRAS  

Lv, Lintong (MOE-LCSM School of Mathematics and Statistics Hunan Normal University)
Yan, Dan (MOE-LCSM School of Mathematics and Statistics Hunan Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.1, 2022 , pp. 73-82 More about this Journal
Abstract
Let K be a field of characteristic zero. We first show that images of the linear derivations and the linear 𝓔-derivations of the polynomial algebra K[x] = K[x1, x2, …, xn] are ideals if the products of any power of eigenvalues of the matrices according to the linear derivations and the linear 𝓔-derivations are not unity. In addition, we prove that the images of D and 𝛿 are Mathieu-Zhao spaces of the polynomial algebra K[x] if D = ∑ni=1 (aixi + bi)∂i and 𝛿 = I - 𝜙, 𝜙(xi) = λixi + 𝜇i for ai, bi, λi, 𝜇i ∈ K for 1 ≤ i ≤ n. Finally, we prove that the image of an affine 𝓔-derivation of the polynomial algebra K[x1, x2] is a Mathieu-Zhao space of the polynomial algebra K[x1, x2]. Hence we give an affirmative answer to the LFED Conjecture for the affine 𝓔-derivations of the polynomial algebra K[x1, x2].
Keywords
LFED conjecture; locally finite ${\varepsilon}$-derivations; Mathieu-Zhao spaces;
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