On free ring extensions of degree n

Abstract

Nagahara and Kishimoto [1] studied free ring extensions B(x) of degree n for some integer n over a ring B with 1, where xn = b, cx = xρ(c) for all c and some b in B(ρ = automophism of B), and {1, x … , xn−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.