Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we set
A = F[ϰ]/(ϰp
Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. ).
In , R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained (,  and ). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer  established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto , among others, have made some progress in case of derivation type.