direct product of R-modules not satisfying the universal property of direct sum I found a case in which the (external) direct sum of R-modules do not satisfy the universal property of the direct product of R-modules.
However, I can't think of one in which the direct product of R-modules do not satisfy the universal property of the direct sum of R-modules....
Could anyone suggest any counter example?

Explicitly: consider the countable product of $\mathbb{Z}$, $\Pi_{i \in \mathbb{N}} \mathbb{Z}$, as a $\mathbb{Z}-$module. Let $\phi_i:\mathbb{Z} \rightarrow \mathbb{Z}$ be the identity map. If the product were to satisfy the universal property of direct sum, there would be a unique map $\phi:\Pi_{i \in \mathbb{N}}\mathbb{Z} \rightarrow \mathbb{Z}$ which is the identity on the coordinates. But it is not true that there is a unique map. For instance, there is no restriction on the value of $\phi(1,1,\ldots,1,\ldots)$
