On H-Skew Polynomial Rings and Galois Extensions

The notion of Azumaya algebras has been generalized to H-separable extensions of noncommutative rings by Hirata (1968). Many properties of H-separable extensions have been discovered by Ikehata (1981,1984), Okamoto (1988) and Sugano (1975). In particular, the set of H-separable skew polynomials of automorphism and derivation types were given by one of the present authors (1981 and 1984), and the set of Hopf Galois H-separable polynomials of derivation type were also obtained by Nakajima (1987). The purpose of the present paper is to investigate H-separable skew polynomial rings of finite degree over a center-Galois extension (that is, the center of the coefficient ring is a Galois extension with Galois group generated by the restriction of ρ). We shall characterize an H-separable skew polynomial ring B[x,ρ] over a center-Galois extension B in terms of the commutator subrings of B and the center Z of B in B[x, ρ]. A further characterization is also obtained when u (= x^m) is in U(Z^ρ) (= the group of units in Z^ρ). Moreover, we shall characterize an H-separable skew polynomial ring of finite degree over an Azumaya algebra B, generalizing Theorem 2.1 in ([I 1]). Furthermore, by using a theorem of DeMeyer (1965), a structure theorem is proved for B[x,ρ] when the Kanzaki hypothesis is satisfied, namely, (1) B is Azumaya 114over Z and (2) Z is Galois over Z^ρ with Galois group <ρ/Z> of order m isomorphic to <ρ>.