We know that tensor product commutes with direct sum. I am wondering if tensor product commutes with sum (or finite sum.)
I.e.) $N\bigotimes (\sum M_i)=\sum(N\otimes M_i)$
$N,M_i$ are $A$ module. ($A$ is commutative ring with unit)
I couldn't find a counter example. Please lead me to find one (if the statement is wrong).
Yes. Tensor products are left adjoint to taking homs, and so they commute with arbitrary colimits. In particular
$$N \otimes \left ( \sum M_\alpha \right ) = \sum (N \otimes M_\alpha)$$
I'm sure there is a more pedestrian way to see this as well, but as some pople say: "category theory exists to make obvious things obviously obvious". I think this is a good example of that principle in action!
After searching, I found a previous question that avoids abstract nonsense.
I hope this helps ^_^