Question about abelian groups, $\mathbb{Z}$-modules and homomorphisms.

Question

I am interested in proving the following issue:

Let $M,N$ be abelian groups then every $\mathbb{Z}$-module homomorphism $f:M\longrightarrow N$ is a homomorphism of abelian groups.

Answer 1

The $\mathbb Z$-module structure on $M$ agrees with the abelian group structure on $M$ and extends it by scalar multiplication $$nx = x + \dots + x, \\-nx = -x-\dots -x$$ for $x \in M$ and $n\in \Bbb N$. Since homomorphisms preserve structure, homomorphisms of $\mathbb Z$-modules in particular preserve the abelian group structure - in fact, this is true for any module.

What is not true in all modules is the converse: Using the definition of scalar multiplication above, every homomorphism of abelian groups is a homomorphism of $\mathbb Z$-modules. This is why it is frequently stated that studying abelian groups is the same as studying $\mathbb Z$-modules.