Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2021.01.11
  • Accepted : 2021.11.05
  • Published : 2022.01.31
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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].

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Acknowledgement

The second author is very grateful to professor Wenhua Zhao for personal communications about the Mathieu-Zhao spaces. She is also grateful to the Department of Mathematics of Illinois State University, where this paper was partially finished, for hospitality during her stay as a visiting scholar. The authors are very grateful to the referee for some useful suggestions.

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