Problem 5. Let $R$ be a commutative ring and let $A$, $B$ and $M$ be $R$-modules. Prove that
$\text{Hom}_R(A \oplus B, M) \cong \text{Hom}_R(A, M) \oplus \text{Hom}_R(B, M)$
and
$\text{Hom}_R(M, A \oplus B) \cong \text{Hom}_R(M, A) \oplus \text{Hom}_R(M, B)$
as $R$-modules.
Hint: This problem is greatly simplified by writing out definitions. If you have a map
$\phi: M \to A \oplus B$, is there a way to break this down into a pair of maps $\phi_1, \phi_2$ where
$\phi_1: M \to A$ and $\phi_2: M \to B$? If so, you've almost proved the second isomorphism.
