Images of Linear Derivations and Linear E-derivations of K[x 1 ,x 2 ,x 3 ]

Abstract: Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED conjecture for linear K-derivations and linear K-E-derivations of K[x 1 ,x 2 ,x 3 ].