On separable extensions of group rings and quaternion rings

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 R G ( 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 R Q 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 R G 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 R Q , and that R Q is separable if and only if 2 is invertible in R .