Proposition 8

a) The homomorphism $\theta_G$ is an isomorphism from $K\otimes_{\mathbf{Z}}R_K(G)$ to $\mathscr{Z}_K(G)$.

b) Suppose that K has characteristic 0. Then $\theta_G$ defines an isomorphism from $R_K(G)$ to the subring of $\mathscr{Z}_K(G)$ consisting of the linear combinations of the characters $\chi_{\lambda}$ for $\lambda$ in $\widehat{G}$ with integer coefficients.

The family $([\lambda ])_{\lambda\in\widehat{G}}$ is a basis of the $\mathbf{Z}$-module $R_K$(G), and the family $(\chi_{\lambda})_{\lambda\in\widehat{G}}$ is a basis of the K-vector space $\mathscr{Z}_K(G)$. Assertions a) and b) follow (VIII, p. 411, Proposition 5).