Let $\{M_i\}_{i \in I}$ and N be left R modules where R is not necessarily commutative. Then how can we prove that
$Hom_R(N, \bigoplus_{i \in I} M_i)$ is isomorphic to $\bigoplus_{i \in I}Hom_R(N,M_i)$.
If I start from $f \in Hom_R(N, \bigoplus_{i \in I} M_i)$ define a map $f_i:N \to M_i$ by $f_i= \pi_i \circ f $ where $\pi_i$ is the projection. Then define $g= \sum_i f_i$. How can we conclude that $g \in \bigoplus_{i \in I}Hom_R(N,M_i)$ ?? How can we prove that all but a finitely many $f_i$ are zero? please help me.
It's not true when $I$ is infinite, due exactly to the problem you encounter. Consider the case where $R = \mathbb{R}$, $N$ is an infinite dimensional vector space, and each $M_i$ is $\mathbb{R}$, with $I$ infinite. Convince yourself that $\operatorname{Hom}(N,\bigoplus_i M_i)$ corresponds to infinite matrices whose columns each have finitely many nonzero entries, while $\bigoplus_i \operatorname{Hom}(N,M_i)$ corresponds to infinite matrices with only a finite number of nonzero rows.
It may be interesting for your purposes to note that both $\operatorname{Hom}(N,\Pi_i M_i)$ and $\Pi_i \operatorname{Hom}(N,M_i)$ correspond to infinite matrices with no finiteness restrictions.
(I know I haven't given a disproof of isomorphism, opting instead to illustrate why we shouldn't expect it to be true for pedagogical reasons.)