Let $R$ and $S$ be algebras over a commutative ring $k$. If $M$ is a left modules over $R$ and $N$ is a left module over $S$ then tensor product $M \otimes_k N$ is a bimodule i.e. left $R \otimes_k S$ module.
Under which assumptions on $R$ and $S$ every every finitely generated $R-S-$bimodule is of the form $$ \oplus_{i=1}^n M_i \otimes_k N_i, $$ where $M_i$ are finitely generated modules over $R$ and $N_i$ are finitely generated over $S$?
If it helps I can assume that $k$ is a field, and $R$ and $S$ are finite dimensional algebras $\dim_k R \leq \infty$, $\dim_k S \leq \infty$ of finite global dimensions $\operatorname{gl}(R) \leq \infty$ and $\operatorname{gl}(S) \leq \infty$