Let K be a field of characteristic zero. We first show that images of the linear derivations and the linear epsilon-derivations of the polynomial algebra K[x] = K[x(1),x(2), ..., x(n)] are ideals if the products of any power of eigenvalues of the matrices according to the linear derivations and the linear epsilon-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 = Sigma(n)(i=1)(a(i)x(i) + b(i))partial derivative(i) and delta = I - phi, phi(x(i)) = lambda(i)x(i) + mu(i) for a(i), b(i), lambda(i), mu(i) is an element of K for 1 <= i <= n. Finally, we prove that the image of an affine 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 epsilon-derivations of the polynomial algebra K[x(1),x(2)].
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