On free ring extensions of degree n

Nagahara and Kishimoto [1] studied free ring extensions B ( x ) of degree n for some integer n over a ring B with 1, where x n = b , c x = x ρ ( c ) for all c and some b in B ( ρ = automophism of B ) , and { 1 , x … , x n − 1 } is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in which b = − 1 and ρ is of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degree n in terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals of B under ρ and the set of ideals of B ( x ) leads to a relation of the Galois extension B over an invariant subring under ρ to the center of B .