On the connected components of the spectrum of the extended character ring of a finite group

Let R(G) be the character ring of a finite group G. For any subring S of the complex field, let π be the set of such rational primes whose inverse do not lie in S, and consider the prime spectrum of the coefficient extended character ring S R(G). By means of a generalized version of Brauerʹs induction theorem, we show that the number of the connected components of the prime spectrum of S R(G) equals the number of the π-regular conjugacy classes of G.