International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 433-438
doi:10.1155/S0161171278000435
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.
Abstract
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 RG(R may be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension RQ over a ring R, where Q are the usual quaternions i,j,k and multiplication and addition are defined as quaternion algebras over a field. We shall show that RG has a unique separable idempotent if and only if G is abelian, that there are more than one separable idempotents for a separable quaternion ring RQ, and that RQ is separable if and only if 2 is invertible in R.