Let $G$ and $H$ be groups, and $R$ a commutative ring. Then elements of $RG$ look like finite sums $\sum\limits_{g\in G}r_g\,g$, and similarly for $RH$. So $RG$ and $RH$ are $R$-modules with bases $G$ and $H$, respectively.
Does it follow that $RG\otimes_R RH$ has basis given by simple tensors $g\otimes h$?