Suppose $G$ and $H$ are discrete groups. If $\rho_G$ and $\rho_H$ are reps of $G$ and $H$ on $V_G$ and $V_H$, respectively, then we get a rep of $G\times H$ on $V_G\otimes V_H$ by sending $(g,h)$ to $\rho_G(g)\otimes\rho_H(h)$. This induces a map $$ R(G) \otimes R(H) \to R(G\times H) $$ where $R(\cdot)$ denotes the representation ring. I know, by character theory, that this is an isomorphism for $G$ and $H$ finite.
I am guessing this result is not true in general-- what is a simple counterexample?
Are there nice conditions on $G$ and $H$ under which the map above is an isomorphism?
Because of some of the issues with the correct definition of representations rings, I will rephrase my question to precisely what I want to know:
Is it always the case that any finite dimensional representation (over $\mathbb C$) of $G \times H$ is isomorphic to $\bigoplus_j V_j \otimes W_j$ where $V_j$ is a rep of $G$ and $W_j$ is a rep of $H$?