example of hom of direct sum I have a question . Can anyone give me examples for $Hom(B,\oplus A_j)$ not isomorphic to $\oplus Hom(B,A_j)$ or $\prod Hom(B,A_j)$ as abelian groups? Here $A_j$ and B are both modules. I have read $Hom(B,\prod A_j)$ is isomorphic to $\prod Hom(B,A_j)$. Thanks for any hint!
 A: There is one example in Rotman's book.
Let $p$ be a prime and $A_n$ a cyclic group of order $p^n$, let
$$
B=\oplus_{n=1}^{\infty}A_n.
$$
Then $\text{Hom}(B,\oplus_{n=1}^{\infty}A_n)$ is not isomorphic to $\oplus_{n=1}^{\infty}\text{Hom}(B,A_n)$. (There is an element of infinite order in $\text{Hom}(B,B)$, but every element in $\oplus_{n=1}^{\infty}\text{Hom}(B,A_n)$ has finite order.)
A: To be able to state generally that $\mathrm{Hom}(M, \oplus N_i) \simeq \oplus \mathrm{Hom}(M, N_i)$, you want the canonical homomorphism $\oplus \mathrm{Hom}(M,N_i) \to \mathrm{Hom}(M,\oplus N_i)$ (which essentially just sums your maps) to be an isomorphism, otherwise any map you can think of which is an isomorphism would rely on the context and is not general. The canonical map above is not always an isomorphism ; consider for instance a module $M$ which possesses a strictly increasing chain of proper submodules, say
$$
M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n \subsetneq \cdots \subseteq M.
$$
Then the map $\oplus \mathrm{Hom}(M, M/M_i) \to \mathrm{Hom}(M, \oplus M/M_i)$ is not surjective since there is a canonical morphism $M \to \oplus M/M_i$ (sending $a$ to $(a+M_0) \oplus (a+M_1) \oplus \cdots \oplus (a+M_n) \oplus \cdots$) which is not the direct sum of finitely many maps $M \to M/M_i$. Of course, this doesn't prove that $\oplus \mathrm{Hom}(M,M/M_i)$ is not isomorphic to $\mathrm{Hom}(M,\oplus M/M_i)$, but an isomorphism between them has to be "unnatural". 
Hope that helps,
