Let $R$ be a ring. $M_R= \oplus_{\alpha \in A}M_{\alpha}$ and $_RN=\oplus _{\beta\in B}N_{\beta}$, then we know that $M \otimes_R N =\oplus_{\alpha \in A, \beta \in B}(M_{\alpha}\otimes_R N_{\beta} )$.
Now suppose that $M_R =\Pi_{\alpha \in A}M_{\alpha}$ and $_RN=\Pi _{\beta\in B}N_{\beta}$(M is the direct product of $M_{\alpha}$ and $N$ is the direct product of $N_{\beta}$), then could we get that $M \otimes_R N =\Pi_{\alpha \in A, \beta \in B}(M_{\alpha}\otimes_R N_{\beta} )$?
