TITLE: Examples proving why the tensor product does not distribute over direct products.
QUESTION [27 upvotes]: I recently read about the result that the tensor product distributes over direct sums. I was curious if it also distributes over direct products, but google tells me it doesn't. 
What are some simple counterexamples to why this property isn't true? I know that there is a natural homomorphism 
$$
\left(\prod M_i\right)\otimes N\to \prod (M_i\otimes N)
$$
given by $(\prod m_i)\otimes n\mapsto \prod (m_i\otimes n)$ when $M$ and $N$ are modules over some commutative ring $R$. Are there standard examples where this homomorphism is not injective/surjective and hence not an isomorphism?

REPLY [7 votes]: Let $X$ and $Y$ be indeterminates. 
For an example where your map is not surjective, take $M_i:=R$ for all $i\in\mathbb N$, and $N:=R[Y]$. 
Then you get the natural map 
$$
R[[X]][Y]\to R[Y][[X]],
$$
and 
$$
\sum_{i\in\mathbb N}\ X^i\ Y^i
$$
is not in the image.
EDIT. Same example with different notation: Put 
$$
A:=\left(\prod M_i\right)\otimes N,\quad B:=\prod\ (M_i\otimes N),
$$
and, for all $i,j\in\mathbb N$, 
$$
M_i=R_i=R_j=R_{ij}=R.
$$
Set also $N:=\bigoplus R_j$. Then we have canonical isomorphisms
$$
A=\bigoplus_j\ \prod_i\ R_{ij},\quad B=\prod_i\ \bigoplus_j\ R_{ij}. 
$$
We also have the inclusions
$$
A\subset B\subset\prod_{i,j}\ R_{ij},
$$
and your map becomes the first inclusion. 
Note that the Kronecker symbol $(\delta_{ij})$ is in $B$ but not in $A$.