Understand homomorphisms from Z^a --> Z^b

[PsychonautQQ]

I want to understand all possible homomorphisms $\alpha: Z^a -> Z^b$ as well as understand what a matrix representation for an arbitrary one of these homomorphisms would look like. Furthermore, under what conditions does a homomorphism have a matrix representation?

To begin, let $(e_1,e_2,...,e_a)$ be a bases for $Z^a$ and $(f_1,f_2,...,f_b)$ be a basis for $Z^b$.

And eh, yeah, can someone give me some insights into my inquiries? Thanks!

 

[fresh_42]

Ring, group or $\mathbb{Z}-$modules? The elements $(0,\ldots,1,\ldots)$ are the generators, so how about defining what happens to them? As $\mathbb{Z}-$modules this isn't very different from ordinary vector space transformations, just with integer matrices.