If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index).
I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly of R-modules, then is there necesarilyl an isomorphism of R-modules $\underset{i\in I}{\prod} M_i \rightarrow \underset{i\in I}{\bigoplus} M_i$?
No, just think for a moment along the lines of $M=\oplus_{i\in \Bbb N} \Bbb Q$ and $N=\prod_{i\in \Bbb N} \Bbb Q$.
The first is countable (being a countable union of countable subsets), and the second is uncountable (you can just adapt Cantor's diagonal argument.)