Example of module homomorphism?

Question

I would like to come up with an example for a module homomorphism between two finitely generated, not free modules over $\mathbb{Z}[i]$. Also, what would be the kernel and image in this example?

Answer 1

How about the identity map $M\to M$, where $M$ is any non-free, finitely generated $\mathbb{Z}[i]$-module?

Do you know how to construct such a thing? Hint: It may be easier to first consider $\mathbb{Z}$.