Given a non-commutative ring $R$, a right module $M$, and a left module $N$, we can define the tensor product $M \otimes_R N$. I suspect that $$ \left(\bigoplus_{i=1}^n M_i\right) \otimes_R N \simeq \bigoplus_{i=1}^n (M_i \otimes_R N) \tag{1} \label{eq:1} $$ if and only if $M_1,\dotsc,M_n,N$ are all bimodules, possibly with the added hypothesis that $R$ is unital, but so far I have been able to prove it. Is this actually true? If not, are there other results on the distributivity of the tensor product for non-commutative rings?
Note: This problem is motivated by this question. In particular, the fact that \eqref{eq:1} holds for modules over a commutative ring seems to suggest that it might reasonably do so for bimodules over a non-commutative ring, at least when it is unital. On the other hand, unless I am mistaken, these simple computations give an example where \eqref{eq:1} fails to hold when $N$ isn't bilateral.
The tensor product of a right module and a left module distributes over all colimits in both variables; you can show this by exhibiting suitable adjunctions. There is no need to assume that anything is a bimodule (which in any case is a structure, not a property). (For me all rings are unital.)