I've been trying to find concrete examples to prove that these statements aren't true:
Let $B$ be an $R$-module and $(A_{j})$ a family of $R$-modules then:
$ \mathrm{Hom}\left ( \prod A_{j} ,B\right )\cong \prod\mathrm{Hom}\left ( A_{j} ,B\right )$
$ \mathrm{Hom}\left ( \prod A_{j} ,B\right )\cong \ \bigoplus\mathrm{Hom}\left ( A_{j} ,B\right )$
For the first statement, I know that if we change the direct product with direct sum in the left side, then it is true. What I've been trying to do is to find an example where the left side is different from zero and when computing the right side, the direct product of Hom's, we get zeros, but I have had no success.
Specifically, I've been struggling to find an example where the left side is nonzero.
Any idea in both cases would be appreciated.