On separable and \(H\)-separable polynomials in skew polynomial rings of several variables

Abstract

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 \(B\).