Gauss extensions and total graded subrings for crossed product algebras

Let V be a total subring of a skew field K and let (G,P) be a right ordered group with P the cone of non-negative elements so that a crossed product K*G of G over K has a right quotient skew field Q(K*G). We want to determine total subrings R of Q(K*G) with R∩K=V, that is, extensions of V in Q(K*G). We describe the class of all those extensions R, called Gauss extensions of V, for which A=R∩K*G is a graded subring of K*G with if for x∈G and a∈K. This can be applied to give explicit constructions of such subrings A and their corresponding extensions R obtained through localization. Information about the prime ideals of R and the graded prime ideals of A is obtained, and it is shown that the skew fields are quotient skew fields of the crossed products for certain subgroups E of G with the residue skew field V/J(V) of V. This result is one of the motivations to consider crossed products K*G rather than just the skew group ring of G over K.