# Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true.

(in other word for every question Do Examples separately (Exemplification)

$${Q1: \operatorname{Hom}_R\left ( \prod M_{i} ,N\right )\not \cong_{\Bbb Z} \bigoplus_{i\in I} \operatorname{Hom}_R\left ( M_{i} ,N\right )}$$
$${Q2: \operatorname{Hom}_R\left ( \prod M_{i} ,N\right )\not \cong_{\Bbb Z} \prod_{i\in I} Hom_R\left ( M_{i} ,N\right )}$$

$${Q3: \operatorname{Hom}_R\left ( M ,\bigoplus N_{i}\right )\not \cong_{\Bbb Z} \bigoplus_{i\in I} \operatorname{Hom}_R\left ( M ,N_{i}\right )}$$

$${Q4:\operatorname{Hom}_R\left ( M ,\bigoplus N_{i}\right )\not \cong_{\Bbb Z} \prod_{i\in I} \operatorname{Hom}_R\left ( M ,N_{i}\right )}$$

in other word this question is four separate question but similar(may be in one answer is existed for all) . i can’t any answer for every case(Q1-Q4).
if you can help me or hint until to aid to solve them (or one)
for this question we just $\Bbb Z_p, \Bbb Q, \Bbb R, \Bbb C$ $R$-modules. and I think we must work with this $R$-modules. but is not work for example (for Q1):

I let $M_i =\Bbb Z_{p_i} , N=\Bbb R$ then $\operatorname{Hom}_R(\prod \Bbb Z_{p_i}, \Bbb R)=\{ \phi \mid \phi : \prod \Bbb Z_{p_i} \rightarrow \Bbb R$ is $R$-homomorphism $\}$

$\operatorname{Hom}_R\left ( \prod M_{i} ,N\right )=\{ \phi \mid \phi : \prod M_{i} \rightarrow N$ is $R$-homomorphism $\}$

$\operatorname{Hom}_R\left ( M_{i} ,N\right )=\{ \phi \mid \phi : M_{i} \rightarrow N$ is $R$-homomorphism $\}$

$\cong_{\Bbb Z}$ is $\Bbb Z$-isomorphism.

$\bigoplus M$ in this statement means submodules of product. In other words it’s product word such that

${“””\bigoplus_{i\in I} M_i=\{ (a_i) \in \prod_{i \in I}M_i : a_i=0}$ for all $i \in I$ except finitely elements $\}”””$

I cannot find the true notation for this operation in help center, so i define this operation under question and in $”””,”””$in fact $\bigoplus$ in this statement is not true and just is used for notation in this way definition.
I must be find for every i to iv statement, examples to show…