This is a question on something on something more general, but for now I'd like to keep in simple. Consider a module homomorphism $\phi:\mathbb{Z}^n\to\mathbb{Z}$, where $n$ is a positive integer. Here I will consider $\mathbb{Z}^n$ as a (left) $\mathbb{Z}$-module. My question is: what are all the types of (module) homomorphisms possible? This question arrises since I was first reading about ring homomorphisms of $\psi:\mathbb{Z}^n\to\mathbb{Z}$ and it seems to me (although I haven't been able to prove it yet) that all said homomorphisms $\psi$ are just projections of the $i$-th (i=1,2,...,n) coordinate (except when $n=1$, then we have the zero homomorphism too). I have a suspicion that all the module homomorphisms aren't going to be of this same manner as the ring homomorphisms, but I might be mistaken.
Thanks for any help in advanced!
$\mathbb Z^n$ is the direct sum of $n$ copies of $\mathbb Z$. The restriction of $\phi$ to each summand induces a module homomorphism from $\mathbb Z$ to $\mathbb Z$, and together these homomorphisms determine $\phi$. The module homomorphisms from $\mathbb Z$ to $\mathbb Z$, in turn, are determined by their value at $1$, and are thus of the form $f_m:k\mapsto km$ with $m\in\mathbb Z$. Thus, the module homomorphisms from $\mathbb Z^n$ to $\mathbb Z$ are the functions $f_x:y\to x\cdot y$ with $x\in\mathbb Z^n$. You can check that $kf_x(y)=k(x\cdot y)=x\cdot(ky)=f_x(ky)$ and $f_x(y+z)=x\cdot(y+z)=x\cdot y+x\cdot z=f_x(y)+f_x(z)$, as required.
This is analogous to the vector space homomorphisms (i.e. linear maps) from $\mathbb R^n$ to $\mathbb R$ being given by $f_x:y\mapsto x\cdot y$ with $x\in\mathbb R^n$.