George Szeto

Copyright © 1981 George Szeto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.