The set of homomorphisms of modules is a module

Question

Suppose $M,N$ are $R$-modules. Let $Hom_R(M,N)$ be the set of all $R$-module homomorphisms from $M$ to $N$ with operations defined by $(\phi+\psi)(m)=\phi(m)+\psi(m)$ and $(r\phi)(m)=r(\phi(m))$. How can I prove that $Hom_R(M,N)$ is an $R$-module?

I just started studying modules and have been very confused, so explanations would be really appreciated!

Answer 1

just prove these for all r , s $\in$ R and $\phi , \psi $ in $*Hom_R(M,N)*$

1. $Hom_R(M,N)$ is an abelian group under the addition $

2. $r(\phi + \psi)$ = $r.\phi +r.\psi$.

3. $(r+s).\phi = r.\phi +s.\phi$

4. $(rs).\phi = r.(s.\phi)$

5. $ 1_R. \phi = \phi$ for all $phi$ in R