International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 433-438
On separable extensions of group rings and quaternion rings
Mathematics Department, Bradley University, Peoria 61625, Illinois, USA
Received 29 November 1977
Copyright © 1978 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.
The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extension ( may be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension over a ring , where are the usual quaternions and multiplication and addition are defined as quaternion algebras over a field. We shall show that has a unique separable idempotent if and only if is abelian, that there are more than one separable idempotents for a separable quaternion ring , and that is separable if and only if is invertible in .