Prove isomorphism for a commutative ring

Question

Show that $M\simeq \operatorname{Hom}_R(R,M)$ considering them as $R$-modules and $R\simeq \operatorname{Hom}_R(R,R)$ considering them as rings.

I couldn't find a way to define the isomorphism.

Answer 1

Hint: It is enough that you define $\phi(1)$. Since $\phi$ is $R$-module homomorphism, we can extent to $R$. Hence $\Phi:\rm{Hom}$$(R,M)\rightarrow M$ where $\Phi$, carries $\phi$ to $\phi(1)$.

Answer 2

Verify that $\hom(R,M) \to M, \phi \mapsto \phi(1)$ is a map, that $M \to \hom(R,M), m \mapsto (r \mapsto rm)$ is a map, and that they are inverse to each other.