On separable group rings

Abstract

Let \(G\) be a finite non-abelian group, \(R\) a ring with 1, and \(\overline G\) the inner automorphism group of the group ring \(RG\) over \(R\) induced by the elements of \(G\). Then three main results are shown for the separable group ring \(RG\) over \(R\): (i) \(RG\) is not a Galois extension of \((RG)^{\overline G}\) with Galois group \(\overline G\) when the order of \(G\) is invertible in \(R\), (ii) an equivalent condition for the Galois map from the subgroups \(H\) of \(G\) to \((RG)^{H}\) by the conjugate action of elements in \(H\) on \(RG\) is given to be one-to-one and for a separable subalgebra of \(RG\) having a preimage, respectively, and (iii) the Galois map is not an onto map.