Universal property of the direct product I want to show that the universal property of the direct product is not satisfied by the direct sum. That means I want to find a ring $R$, modules $M_i$ and $N$ and module homomorphisms $f_i:N\rightarrow M_i$ such that an $f:N\rightarrow\bigoplus_iM_i$ is not unique or does not exists.
I tried to choose $M_i=R=N=\mathbb{Z}$ but could not prove the claim. What can I choose?
 A: Here's a proof that direct sums and products are not necessarily isomorphic, without using the universal property: Direct sums and direct products are unique up to isomorphism. The direct sum $\oplus_{\mathbb{N}}\mathbb{Z}$ is countable, but the direct product $\prod_{\mathbb{N}}\mathbb{Z}$ is uncountable, so they are not isomorphic.
It is a little strange to try to prove that the universal property of direct sums is not satisfied by direct products, because a direct sum $\bigoplus_i M_i$ comes with embeddings $M_j\hookrightarrow\bigoplus_i M_i$, but a direct product $\prod_i M_i$ comes with projections $\prod_i M_i\twoheadrightarrow M_j$, so the arrows are in the opposite direction.
However, we do know the usual descriptions of direct sums and products, and there are usual embeddings $M_j\to\prod M_i$ (although I wouldn't call them canonical). If we consider these embeddings then we can look at a similar counter-example: Take $\mathbb{Q}$ instead of $\mathbb{Z}$. Take the identity $\mathbb{Q}\to\mathbb{Q}$. By choosing an appropriate basis of $\prod\mathbb{Q}$, we can find two distinct morphisms $p_1,p_2:\prod\mathbb{Q}\to\mathbb{Q}$ for which $\mathbb{Q}\hookrightarrow\prod \mathbb{Q}\overset{p_i}{\to}\mathbb{Q}$ is the identity, so uniqueness is not satisfied.
