On separable and $H$-separable polynomials in skew polynomial rings of several variables

Let $B$ be a ring with 1, and $\{\rho_1,\cdots,\rho_e\}$ a set of automorphisms of $B$. Let $B[X_1, \cdots,X_e;\rho_1,\cdots,\rho_e;\{ u_{ij}\}]$ be the skew polynomial ring of automorphism type. In this paper, we shall give equivalent conditions that the residue ring of $B[X_1,\cdots,X_e;\rho_1,\cdots,\rho_e;\{ u_{ij}\}]$ by the ideal generated by a set $\{X_1^{m_1}-u_1,\cdots,X_e^{m_e}-u_e\}$ to be separable or $H$-separable over $B$.