TITLE: Examples proving that the tensor product does not commute with direct products
QUESTION [9 upvotes]: Examples proving why the tensor product does not distribute over direct products?
In fact the canonical map is not surjective; can you give me a simple example?

REPLY [6 votes]: The canonical map can fail to be injective: if $A$, $B_n$ are $R$-modules, then tensoring the canonical projections $\prod B_n \to B_n$ with $A$ give maps $A \otimes \prod B_n \to A \otimes B_n$, which induce the canonical map $A \otimes \prod B_n \to \prod (A \otimes B_n)$. It can happen that $A \otimes B_n = 0$ for all $n$, but $A \otimes \prod B_n \ne 0$. 
Take $R = \mathbb{Z}$, $A = \mathbb{Q}$, and $B_n = \mathbb{Z}/n\mathbb{Z}$: then $A \otimes B_n = 0$ for all $n$, but $A \otimes \prod B_n = S^{-1} \prod B_n$, where $S = \mathbb{Z} \setminus \{0\}$, and the element $(1, 1, \ldots) \in \prod B_n$ is not annihilated by any element of $S$, so $A \otimes \prod B_n \ne 0$ (since for any $\mathbb{Z}$-module $M$, $S^{-1}M = 0$ iff every element of $M$ is annihilated by some element of $S$).
Notice that this is an example where localizations do not commute with infinite direct products. In particular, even tensoring with a flat module does not commute with direct products.