I am given that $A$ a local ring, $\mathfrak{m}$ its maximal ideal and $M,N$ be finitely generated $A$-modules. $\phi: M \to N$ be homomorphism. I have to show that $\tilde{\phi}: M/\mathfrak{m}M\to N/\mathfrak{m}N$ induces a homomorphism of $A$-modules and that $\phi$ is surjective if and only if $\tilde{\phi}$ is surjective.
For the first part I thought take $\tilde{x}\in M/\mathfrak{m}$ then this corresponds to $x\in M$ such that $\tilde{x}=x+\mathfrak{m}$ so $\tilde{\phi}(\tilde{x})=\phi(x+\mathfrak{m})=\phi(x)+\phi(\mathfrak{m})=y+\mathfrak{m}$ so this corresponds $\tilde{y}$ such that $\tilde{y}=y+\mathfrak{m}$. So it is well-defined and then the properties of homomorphism are just trivial. So this induces a homomorphism. Is this correct or am I just naïve?
The second part I don't really know how to approach
