Direct sum and Hom Let $R$ be a commutative ring with unity and let $M$ be a finitely generated noetherian R-module. 
Can someone tell me how the isomorphism 
$$
Hom_R(R\oplus R, M)\simeq Hom_R(R,M)\oplus Hom_R(R,M)\simeq M\oplus M
$$
is given? I know how the second isomorphism is given but I don't know about the first. Where does it send a map $R\oplus R\to M$ to?

The other answer is nice but only if you're comfortable with the language of categories and universal properties. More concretely, if we have a homomorphism
$$\phi:R\oplus R\to M,$$
we get two homomorphisms $\phi_1,\phi_2:R\to M$ by defining
$$\phi_1(r)=\phi(r,0),\quad\quad\phi_2(r)=\phi(0,r).$$
You can check this sending $\phi\mapsto(\phi_1,\phi_2)$ gives an isomorphism $$\hom(R\oplus R,M)\cong\hom(R,M)\oplus\hom(R,M).$$
