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.

Highlights

  • INTRODUCTIONSzeto ([4] [5])

  • Kishimoto [3], and Nagahara and Kishimoto [i] studied free ring extensions ofG

  • One of their results is a characterization of the Galois extension of B over a subring ([2], Proposition 1.1): Let B(x) be a generalized quaternion algebra

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Summary

INTRODUCTION

Szeto ([4] [5]) One of their results is a characterization of the Galois extension of B over a subring ([2], Proposition 1.1): Let B(x) be a generalized quaternion algebra. The above characterization was generalized to a free ring extension of degree n, B(x) with xn -I We shall generalize the Parimala-Sridharan [2] theorem to a free ring extension B(x) of degree n for an integer n such that xn b and ax xp(a) for some b and all a in B where p is an automorphism of B of order n. By the Parimla-Sridharan theorem ([3], Proposition i.I), let B(x) be a generalized quaternion algebra (x2 =-i) over a commutative ring B.

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