International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 2, Pages 237-245
doi:10.1155/S0161171280000166

On generalized quaternion algebras

Abstract

Let B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of this paper is to study the separability of B[j]. The separable extension of B[j] over B is characterized in terms of the trace (=1+σ) of B over the subring of fixed elements under σ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved.