The Image Of Some Derivations And ?-derivations Over Integral Domain Of Characteristic P

In this paper,we classify polynomial algebras and ideals over integral domain of characteristic p,and study their images under some derivations or?-derivations.We get the following conclusions.K is a field of characteristic p,and R is an integral domain of characteristic p.?=I-? is a one-dimensional K-?-derivation,D=x₁^jf(x₁^p)(?)₁(0?j<p)is a one-dimensional R-derivation.Firstly,we prove that Im?(or ImD)is a Mathieu-Zhao subspace of K[x₁](or R[x₁]),if and only if ?(or D)is not locally nilpotent.Secondly,we study more general R-derivations D=f(x₁)(?)₁,and obtain a necessary and sufficient condition for ImD or DI to be a Mathieu-Zhao subspace of R[x₁].In particular,ImD or DI is a Mathieu-Zhao subspace of R[x₁],if and only if D is not locally nilpotent R-derivation over an integral domain with characteristic 2.And we give some necessary and sufficient conditions for the image of some ideals under K-?-derivation to be Mathieu-Zhao subspace of K[x₁].Finally,on the integral domain of characteristic 2,we study the twodimensional derivation D=r1(x₁,x₂)(?)₁+r2(x₁,x₂)(?)₂.