Let $R(G)$ be the representation ring of $G$ over $\mathbb{C}$ and $\mathbb{C}_{class}(G)=\{\text{class functions on $G$ to $\mathbb{C}$}\}$. There is injective ring homomorphism $$\chi: R(G) \to \mathbb{C}_{class}(G)$$ maps virtual representations to their virtual characters.
The book that I am reading (Fulton&Harris) states that "our last proposition amounts to the statement that $\chi$ induces an isomorphism $$\chi_{\mathbb{C}}: R(G) \otimes \mathbb{C} \to \mathbb{C}_{class}(G),"$$
where the last proposition is the fact that the character of irreps form an orthonormal basis for $\mathbb{C}_{class}(G)$.
I do not understand why their last proposition is equivalent to the mentioned isomorphism. Can you explain why? Furthermore, I have not seen the tensor product of rings. I have only seen the tensor product of modules and that of algebras. So, do the authors consider $R(G)$ as a $\mathbb{C}$-algebra? Is $\chi_{\mathbb{C}}$ isomorphism of algebras?