An alternative proof of Miyashita's theorem in a skew polynomial ring II

Y. Miyashita gave characterizations of a separable polynomial and a Hirata separable polynomial in skew polynomial rings. In the previous paper, the author and S. Ikehata gave direct and elementary proofs of Miyashita’s theorems in skew polynomial rings of automorphism type $B[X;\rho ]$ and derivation type $B[X;D]$, respectively. The purpose of this paper is to give proofs for them in the general skew polynomial ring $B[X;\rho ,D]$.